famous mathematics quotes

8/3/2019 Famous Mathematics Quotes

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If people do not believe that mathematics is simple, it is only because they do notrealize how complicated life is. ~John Louis von Neumann

Arithmetic is where the answer is right and everything is nice and you can look out of

the window and see the blue sky - or the answer is wrong and you have to start over andtry again and see how it comes out this time. ~Carl Sandburg

Pure mathematics is, in its way, the poetry of logical ideas. ~Albert Einstein

Mathematics are well and good but nature keeps dragging us around by the nose.~Albert Einstein

Black holes result from God dividing the universe by zero. ~Author Unknown

Mathematics - the unshaken Foundation of Sciences, and the plentiful Fountain ofAdvantage to human affairs. ~Isaac Barrow

I never did very well in math - I could never seem to persuade the teacher that I hadn'tmeant my answers literally. ~Calvin Trillin

I don't agree with mathematics; the sum total of zeros is a frightening figure.

~Stanislaw J. Lec,More Unkempt Thoughts

If you think dogs can't count, try putting three dog biscuits in your pocket and thengiving Fido only two of them. ~Phil Pastoret

[A mathematician is a] scientist who can figure out anything except such simple thingsas squaring the circle and trisecting an angle. ~Evan Esar,Esar's Comic Dictionary

"Every minute dies a man, Every minute one is born;" I need hardly point out to youthat this calculation would tend to keep the sum total of the world's population in a stateof perpetual equipoise, whereas it is a well-known fact that the said sum total isconstantly on the increase. I would therefore take the liberty of suggesting that in thenext edition of your excellent poem the erroneous calculation to which I refer should becorrected as follows: "Every moment dies a man, And one and a sixteenth is born." Imay add that the exact figures are 1.067, but something must, of course, be conceded tothe laws of metre. ~Charles Babbage, letter to Alfred, Lord Tennyson, about a couplet in


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his "The Vision of Sin"

Math is radical! ~Bumper Sticker

There was a blithe certainty that came from first comprehending the full Einstein fieldequations, arabesques of Greek letters clinging tenuously to the page, a gossamer web.They seemed insubstantial when you first saw them, a string of squiggles. Yet to followthe delicate tensors as they contracted, as the superscripts paired with subscripts,collapsing mathematically into concrete classical entities - potential; mass; forcesvectoring in a curved geometry - that was a sublime experience. The iron fist of the real,inside the velvet glove of airy mathematics. ~Gregory Benford, Timescape

It is a mathematical fact that fifty percent of all doctors graduate in the bottom half oftheir class. ~Author Unknown

If two wrongs don't make a right, try three. ~Author Unknown

Arithmetic is where numbers fly like pigeons in and out of your head. ~Carl Sandburg,"Arithmetic"

Arithmetic is numbers you squeeze from your head to your hand to your pencil to yourpaper till you get the answer. ~Carl Sandburg, "Arithmetic"

If equations are trains threading the landscape of numbers, then no train stops at pi.~Richard Preston

Even stranger things have happened; and perhaps the strangest of all is the marvel thatmathematics should be possible to a race akin to the apes. ~Eric T. Bell,TheDevelopment of Mathematics

So if a man's wit be wandering, let him study the mathematics; for in demonstrations, ifhis wit be called away never so little, he must begin again. ~Francis Bacon, "Of Studies"

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


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The human mind has never invented a labor-saving machine equal to algebra. ~AuthorUnknown

The mathematics are distinguished by a particular privilege, that is, in the course of

ages, they may always advance and can never recede. ~Edward Gibbon,Decline andFall of the Roman Empire

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

It is not the job of mathematicians... to do correct arithmetical operations. It is the jobof bank accountants. ~Samuil Shchatunovski

Trigonometry is a sine of the times. ~Author Unknown

Mathematics is not a careful march down a well-cleared highway, but a journey into astrange wilderness, where the explorers often get lost. Rigour should be a signal to thehistorian that the maps have been made, and the real explorers have gone elsewhere.~W.S. Anglin

A mathematician is a device for turning coffee into theorems. ~Paul Erdos

Let us grant that the pursuit of mathematics is a divine madness of the human spirit, arefuge from the goading urgency of contingent happenings. ~Alfred North Whitehead

The tantalizing and compelling pursuit of mathematical problems offers mentalabsorption, peace of mind amid endless challenges, repose in activity, battle withoutconflict, "refuge from the goading urgency of contingent happenings," and the sort ofbeauty changeless mountains present to sense tried by the present-day kaleidoscope ofevents. ~Morris Kline,Mathematics in Western Culture

Mathematics is as much an aspect of culture as it is a collection of algorithms. ~CarlBoyer, 1949, calculus textbook

The cowboys have a way of trussing up a steer or a pugnacious bronco which fixes thebrute so that it can neither move nor think. This is the hog-tie, and it is what Euclid didto geometry. ~Eric Bell, The Search for Truth


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Sometimes it is useful to know how large your zero is. ~Author Unknown

The hardest arithmetic to master is that which enables us to count our blessings. ~Eric

Hoffer,Reflections On The Human Condition

Mathematics is the only good metaphysics. ~William Thomson Baron Kelvin of Largs

The most savage controversies are those about matters as to which there is no goodevidence either way. Persecution is used in theology, not in arithmetic. ~BertrandRussell

Music is the pleasure the human mind experiences from counting without being awarethat it is counting. ~Gottfried Leibniz

How many times can you subtract 7 from 83, and what is left afterwards? You cansubtract it as many times as you want, and it leaves 76 every time. ~Author Unknown

To most outsiders, modern mathematics is unknown territory. Its borders are protectedby dense thickets of technical terms; its landscapes are a mass of indecipherableequations and incomprehensible concepts. Few realize that the world of modern

mathematics is rich with vivid images and provocative ideas. ~Ivars Peterson

With my full philosophical rucksack I can only climb slowly up the mountain ofmathematics. ~Ludwig Wittgenstein, Culture and Value

But mathematics is the sister, as well as the servant, of the arts and is touched with thesame madness and genius. ~Harold Marston Morse

Still more astonishing is that world of rigorous fantasy we call mathematics. ~GregoryBateson

Anyone who cannot cope with mathematics is not fully human. At best he is a tolerablesubhuman who has learned to wear shoes, bathe, and not make messes in the house.~Robert Heinlein, Time Enough for Love


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You may be an engineer if your idea of good interpersonal communication meansgetting the decimal point in the right place. ~Author Unknown

The trouble with integers is that we have examined only the very small ones. Maybe allthe exciting stuff happens at really big numbers, ones we can't even begin to think aboutin any very definite way. Our brains have evolved to get us out of the rain, find wherethe berries are, and keep us from getting killed. Our brains did not evolve to help usgrasp really large numbers or to look at things in a hundred thousand dimensions.~Ronald L. Graham

We could use up two Eternities in learning all that is to be learned about our own worldand the thousands of nations that have arisen and flourished and vanished from it.Mathematics alone would occupy me eight million years. ~Mark Twain

Can you do Division? Divide a loaf by a knife - what's the answer tothat? ~LewisCarroll, Through the Looking Glass

Mathematics is the supreme judge; from its decisions there is no appeal. ~TobiasDantzig

Although he may not always recognize his bondage, modern man lives under a tyrannyof numbers. ~Nicholas Eberstadt, The Tyranny of Numbers: Mismeasurement and

Misrule

The mathematics are usually considered as being the very antipodes of Poesy. YetMathesis and Poesy are of the closest kindred, for they are both works of theimagination. ~Thomas Hill

The temperature of Heaven can be rather accurately computed. Our authority is Isaiah30:26, "Moreover, the light of the Moon shall be as the light of the Sun and the light ofthe Sun shall be sevenfold, as the light of seven days." Thus Heaven receives from the

Moon as much radiation as we do from the Sun, and in addition 7*7 (49) times as muchas the Earth does from the Sun, or 50 times in all. The light we receive from the Moon isone 1/10,000 of the light we receive from the Sun, so we can ignore that.... Theradiation falling on Heaven will heat it to the point where the heat lost by radiation isjust equal to the heat received by radiation, i.e., Heaven loses 50 times as much heat asthe Earth by radiation. Using the Stefan-Boltzmann law for radiation, (H/E)temperature of the earth (-300K), gives H as 798K (525C). The exact temperature ofHell cannot be computed.... [However] Revelations 21:8 says "But the fearful, and


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unbelieving...shall have their part in the lake which burneth with fire and brimstone." Alake of molten brimstone means that its temperature must be at or below the boilingpoint, 444.6C. We have, then, that Heaven, at 525C is hotter than Hell at 445C. ~FromApplied Optics, vol. 11, A14, 1972

I used to love mathematics for its own sake, and I still do, because it allows for nohypocrisy and no vagueness.... ~Stendhal (Henri Beyle), The Life of Henri Brulard

As far as the laws of mathematics refer to reality, they are not certain; and as far as theyare certain, they do not refer to reality. ~Albert Einstein,Sidelights on Relativity

God does not care about our mathematical difficulties; He integrates empirically.~Albert Einstein

If there is a God, he's a great mathematician. ~Paul Dirac

To all of us who hold the Christian belief that God is truth, anything that is true is a factabout God, and mathematics is a branch of theology. ~Hilda Phoebe Hudson

[T]he different branches of Arithmetic - Ambition, Distraction, Uglification, andDerision. ~Lewis Carroll

[G]eometry is not true, it is advantageous. ~Henri Poincar

Infinity is a floorless room without walls or ceiling. ~Author Unknown

Physics is mathematical not because we know so much about the physical world, butbecause we know so little; it is only its mathematical properties that we can discover.~Bertrand Russell

God is real, unless declared integer. ~Author Unknown

If a healthy minded person takes an interest in science, he gets busy with hismathematics and haunts the laboratory. ~W.S. Franklin


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Proofis an idol before whom the pure mathematician tortures himself. ~Arthur StanleyEddington, The Nature of the Physical World

There was a young man from Trinity,

Who solved the square root of infinity.While counting the digits,He was seized by the fidgets,Dropped science, and took up divinity.~Author Unknown

One cannot escape the feeling that these mathematical formulas have an independentexistence and an intelligence of their own, that they are wiser than we are, wiser eventhan their discoverers... ~Heinrich Hertz

In the binary system we count on our fists instead of on our fingers. ~Author Unknown

There are 10 types of people in this world: those who understand binary and those whodon't. ~Author Unknown

The laws of nature are but the mathematical thoughts of God. ~Euclid

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 tothe old structure. ~Hermann Hankel

Twice two makes four seems to me simply a piece of insolence. Twice two makes four isa pert coxcomb who stands with arms akimbo barring your path and spitting. I admitthat twice two makes four is an excellent thing, but if we are to give everything its due,twice two makes five is sometimes a very charming thing too. ~Fyodor MikhailovichDostoevsky

I know that two and two make four - & should be glad to prove it too if I could - though Imust say if by any sort of process I could convert 2 & 2 intofive it would give me muchgreater pleasure. ~George Gordon, Lord Byron

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


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Nature does not count nor do integers occur in nature. Man made them all, integers andall the rest, Kronecker to the contrary notwithstanding. ~Percy William Bridgman,TheWay Things Are

A man has one hundred dollars and you leave him with two dollars. That's subtraction.~Mae West

I've dealt with numbers all my life, of course, and after a while you begin to feel thateach number has a personality of its own. A twelve is very different from a thirteen, forexample. Twelve is upright, conscientious, intelligent, whereas thirteen is a loner, ashady character who won't think twice about breaking the law to get what he wants.Eleven is tough, an outdoorsman who likes tramping through woods and scalingmountains; ten is rather simpleminded, a bland figure who always does what he's told;nine is deep and mystical, a Buddha of contemplation.... ~Paul Auster,The Music ofChance

Why do we believe that in all matters the odd numbers are more powerful? ~Pliny theElder,Natural History

Uneven numbers are the gods' delight. ~Virgil, The Eclogues

One of the endlessly alluring aspects of mathematics is that its thorniest paradoxes have

a way of blooming into beautiful theories. ~Philip J. Davis

Pure mathematics is the world's best game. It is more absorbing than chess, more of agamble than poker, and lasts longer than Monopoly. It's free. It can be playedanywhere - Archimedes did it in a bathtub. ~Richard J. Trudeau,Dots and Lines

I like mathematics because it is not human and has nothing particular to do with thisplanet or with the whole accidental universe - because like Spinoza's God, it won't loveus in return. ~Bertrand Russell, 1912

The man ignorant of mathematics will be increasingly limited in his grasp of the mainforces of civilization. ~John Kemeny

Although I am almost illiterate mathematically, I grasped very early in life that any onewho can count to ten can count upward indefinitely if he is fool enough to do so.


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~Robertson Davies, "Of the Conservation of Youth," The Table Talk of SamuelMarchbanks

A million thanks to Norm & Andy for submitting some of these quotes!

Math Quotes and Sayings

Math is like love : a simple idea but it can get complicated.

The essence of math is not to make simple things complicated, but to make complicated thingssimple.

Math is the science which uses easy words for hard ideas.

Round numbers are always false.

For the things of this world cannot be made known without a knowledge of math.

In mathematics I can report no deficiency, except that it be that men do not sufficently

understand the use of the Pure Mathematics.

Not everything that can be counted counts, and not everything that counts can be counted.

Mathematics is the handwriting on the human consciousness of the very Spirit of Life itself.

2 is not equal to 3 not even for very large values of 2.

The mathematics is not there till we put it there.

Twelve for 23 It doesnt take a genius to see thats under 50 percent.

Obvious is the most dangerous word in mathematics.

God exists since mathematics is consistent, and the Devil exists since we cannot prove it.

In mathematics you dont understand things. You just get used to them.

What is algebra exactly; is it those three-cornered things ?

One can always reason with reason.

The longer math lives the more abstract and therefore, possibly also the more practical itbecomes.


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It is not of the essence of mathematics to be occupied with the ideas of number and quantity.

I have hardly ever known a mathematician who was capable of reasoning.

Math is the queen of the sciences.

To the extent math refers to reality, we are not certain; to the extent we are certain, math does not

refer to reality.

A thing is obvious mathematically after you see it.

Math may be defined as the subject in which we never know what we are talking about, norwhether what we are saying is true.

I dont believe in mathematics.

Math is the tool specially suited for dealing with abstract concepts of any kind and there is nolimit to its power in this field.

Mighty is geometry; joined with art, resistless.

As for everything else, so for a mathematical theory: beauty can be perceived but not explained.

The essence of mathematics is its freedom.

As far as the laws of mathematics refer to reality, they are not certain; and as far as they arecertain, they do not refer to reality.

It is as if mathematics were the vegetables of the academic dinner: Everyone knows that they aregood for you, but no one forces you to eat them.

Math seems to endow one with something like a new sense.

Pure Mathematics is the science of the individual object in as much as it is born in thought.

Everything existing in the Universe is the fruit of chance and necessity.

Do not worry about your difficulties in Mathematics. I can assure you mine are still greater.

Each problem that I solved became a rule which served afterwards to solve other problems.

Famous Mathematics Quotes

Archimedes of Syracus (287-212 B. C. E)


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y Give me a place to stand, and I will move the earth.y Eureka, euraka!y Don't spoil my circles! (or Do not disturb my circles!)y There are things which seem incredible to most men who have not studied Mathematics.

Aristotle (384-322 B. C. E)y Now what is characteristic of any nature is that which is best for it and gives most joy. Such a

man is the life according to reason, since it is that which makes him man.

y There is nothing strange in the circle being the origin of any and every marvel.y The so-called Pythagoreans, who were the first to take up mathematics, not only advanced this

subject, but saturated with it, they fancied that the principles of mathematics were the

principles of all things.

y To Thales the primary question was not what do we know, but how do we know it.y If this is a straight line [showing his audience a straight line drawn by a ruler], then it necessarily

ensues that the sum of the angles of the triangle is equal to two right angles, and conversely, if

the sum is not equal to two right angles, then neither is the triangle rectilinear.y It is not once nor twice but times without number that the same ideas make their appearance in

the world.

y But Nature flies from the infinite, for the infinite is unending or imperfect, and Nature everseeks an end.

y We cannot ... prove geometrical truths by arithmetic.y The chief forms of beauty are order and symmetry and definiteness, which the mathematical

sciences demonstrate in a special degree.

y The continuum is that which is divisible into indivisibles that are infinitely divisible. Physics.Roger Bacon (1214-1294)

y In mathematics I can report no deficience, except it be that men do not sufficiently understandthe excellent use of the Pure Mathematics.

y Mathematics is the door and key to the sciences.y Neglect of mathematics works injury to all knowledge, since he who is ignorant of it cannot

know the other sciences or the things of the world.

y There are four great sciences ... Of these sciences the gate and key is mathematics, which thesaints discovered at the beginning of the world.

y ... mathematics is absolutely necessary and useful to the other sciences.

Heri Bergson (1859-1941)

y One can always reason with reason.Janos Bolyai (1802-1860)

y I have created a new universe from nothing.y One must do not violence to nature, nor model it in conformity to any blindly formed chimaera.


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Bernhard Bolzano (1781-1848)

y My special pleasure in mathematics rested particularly on its purely speculative part.y Even in the realm of things which do not claim actuality, and do not even claim possibility, there

exist beyond dispute sets which are infinite.

George Boole (1815-1869)

y It is not of the essence of mathematics to be occupied with the ideas of number and quantity.y 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.

Geoge Cantor (1845-1918)

y The essence of mathematics is its freedom.y I see it, but I don't believe it.y Mathematics is entirely free in its development, and its concepts are only linked by the necessity

of being consistent, and are co-ordinated with concepts introduced previously by means of

precise definitions.

y In mathematics the art of proposing a question must be held of higher value than solving it.y every transfinite consistent multiplicity, that is, every transfinite set, must have a definite aleph

as its cardinal number.

Rene Descartes (1596-1650)

y I think, therefore I am.y Perfect numbers like perfect men are very rare.y With me everything turns into mathematics.y It is not enough to have a good mind. The main thing is to use it well.y ... the two operations of our understanding, intuition and deduction, on wh ich alone we have

said we must rely in the acquisition of knowledge.

y In order to seek truth it is necessary once in the course of our life to doubt as far as possible allthings.

y Give me extension and motion and I will construct the universe.y There have been only Mathematicians who were able to find some proofs, that is to say some

sure and certain reasons.

y Take what you need; act as you must, and you will obtain that for which you wish!y Only having one truth about each object, whoever finds it knows as much as can known about it.

Democritus (460-370 B. C)

y Found, but not proven.y No one has ever surpassed me in constructing figures by means of proofs, not even the Egyptian

``harpedouaptes''(knotters of ropes or geometry), as they are called.


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y I would rather discover one scientific fact than become King of Persia.y Everything existing in the Universe is the fruit of chance and necessity.y Nothing exists except atoms and empty space; everything else is opinion.

Albert Einstein (1879-1955)

y So far as the theories of mathematics are about reality, they are not certain; so far as they arecertain, they are not about reality.

y I don't believe in mathematics.y God does not care about our mathematical difficulties. He integrates empirically.y Nature to him (Newton) was an open book, whose letters he could read without effort.y Since the mathematicians have invaded the theory of relativity, I do not understand it myself

anymore.

y Do not worry about your difficulties in mathematics, I assure you that mine are greater.

Euclid of Alexandria (325-265 B.C.E)

y There is no royal road to geometry.y A youth who had begun to read geometry with Euclid, when he had learnt the first proposition,

inquired, "What do I get by learning these things?" So Euclid called a slave and said "Give him

threepence, since he must make a gain out of what he learns."

Eudoxus of Cnidus (408-355 B. C. E)

y Willingly would I burn to death like Phaeton, were this the price for reaching the sun andlearning its shape, its size and its substance.

Euripides (485-406 B. C. E)

y Mighty is geometry; joined with art, resistless.ierre de Fermat (1601-1665)

y To divide a cube into two other cubes, a fourth power or in general any power whatever intotwo powers of the same denomination above the second is impossible, and I have assuredly

found an admirable proof of this, but the margin is too narrow to contain it.

y And perhaps, posterity will thank me for having shown it that the ancients did not knoweverything.

Friedrich Ludwig Gottlob Frege (1848-1925)

y Arithmetic has began to totter.y Every good mathematician is at least half a philosopher, and every good philosopher is at least

half a mathematician.


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y What are numbers?What is the nature of arithmetical truth?Carl Friedrich Gauss (1777-1855)-German

y Thou, nature, art my goddess; to thy laws My Services are bound.y I am ever more convinced that the necessity of our geometry cannot be proved -- at least not by

human reason for human reason.

y One sees by that, what should be understood by a class composed from two or more class.y Mathematics is the queen of the sciences and number theory is the queen of mathematics.y The total number ofDirichlet's publications is not large: jewels are not weighed on a grocery

scale.

y I confess that Fermat's Theorem as an isolated proposition has very little interest for me,because I could easily lay down a multitude of such propositions, which one could neither prove

nor dispose of.

y God does arithmetic.y I have had my results for a long time: but I do not yet know how I am to arrive at them.y Mathematics is concerned only with the enumeration and comparison of relations.

Kurt Godel (1906-1978)

y Nothing new had been done in Logic since Aristotle!y Either mathematics is too big for the human mind or the human mind is more than a machine.y I don't believe in natural science.y 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.

y ...a consistency proof for [any] system ... can be carried out only by means of modes of inferencethat are not formalized in the system ... itself.

aul Albert Gordan (1837-1912)

y This (axiomatic math) is no longer mathematics, it is theology.Hermann Gunter Grassmann (1809-1877)

y Pure Mathematics is the science of the individual object in as much as it is born in thought.y Pure Mathematics is the theory of forms.

Jacques Salomon Hadamard (1865-1963)

y Can the existence of a mathematical entity be proved without definiing it ?y The shortest path between two truths in the real domain passes through the complex domain.y Practical application is found by not looking for it, and one can say that the whole progress of

civilization rests on that principle.


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y To parents who despair because their children are unable to master the first problems inarithmetic I can dedicate my examples. For, in arithmetic, until the seventh grade I was last or

nearly last.

y Logic merely sanctions the conquests of the intuition.

Hermann Hankel (1839-1873)y purely intellectual, a pure theory of forms, which has as its purpose, not the combination of

quantities, or of their images, the numbers, but objects of thought to which may correspond

effectives or relations, even though such a correspondence is not necessary.

y In most sciences one generation tears down what another has built, and what one hasestablished, another undoes. In Mathematics alone each generation adds a new storey to the

old structure.

Charles Hermite (1822-1901)

y Turn away with fear and horror from this lamentable plague of continuous functions that do nothave a derivative.

y I believe that numbers and functions ofAnalysis are not the arbitrary result of our minds; I thinkthat they exist outside ofus, with the same character of necessity as the things of objective

reality, and we meet them or discover them, and stuty them, as do the physicists, the chemists

and the zoologists.

y There exists, if I am not mistaken, an entire world which is the totality of mathematical truths, towhich 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.

y We are servants rather than masters in mathematics.y Analysis takes back with one hand what it gives with the other. I recoil in fear and loathing from

that deplorable evil: continuous functions with no derivatives.

David Hilbert (1862-1943)

y We must know, we will know.y Let us consider three distinct systems of things. (refer to Chair, table, beer-mug as point, line

and plane for axioms)

y Mathematics is a game played according to certain rules with meaningless marks on paper.y No one will expel us from this paradise Cantor has created for us.y The finest product (Cantor's work on set theory) of mathematical genius and one of the

supreme achievments of purly intellectual human activity.

y I have tried to avoid long numerical computations, thereby following Riemann's postulate thatproofs should be given through ideas and not voluminous computations.

y The art of doing mathematics consists in finding that special case which contains all the germs ofgenerality.

y Mathematics knows no races or geographic boundaries; for mathematics, the cultural world isone country.

y The infinite! No other question has ever moved so profoundly the spirit of man.


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y The faculty is not a pool changing room. [On the proposed appointment of the first womanprofessor.]

y If I were to awaken after having slept for a thousand years, my first question would be: Has theRiemann hypothesis been proven?

Carl Gustav Jacobi (1804-1851)y It is a pity that you (Gauss) did not publish this result, since you have published so many poorer

papers.

y It will not be possible to elucidate it (refers to Grammer rule for degenerate coefficent matrix)briefly.

y God ever arithmetizes.y One should always generalize.y The real end of science is the honour of the human mind.y Mathematics is the science of what is clear by itself.y The God that reigns in Olympus is Number Eternal.

Lord William Thomson Kelvin (1824-1907)

y When you can measure what you are talking about and express it in numbers, you knowsomething about it.

Johannes Kepler (1571-1630)

y However, before we come to [special] creation, which puts an end to all discussion: I think weshould try everything else.

yGeometry is one and eternal shining in the mind of God.

T

hat share in it accorded to men is oneof the reasons that Man is the image of God.

y Truth is the daughter of time, and I feel no shame in being her midwife.y ...I am stealing the golden vessels of the Egyptians to build a tabernacle to my God from them,

far far away from the boundaries ofEgypt. If you forgive me, I shall rejoice.; if you are enraged

with me, I shall bear it. See, I cast the die, and I write the book. Whether it is to be read by the

people of the present or of the future makes no difference: let it await its reader for a hundred

years, if God himself has stood ready for six thousand years for one to study him.

y I used to measure the Heavens, now I measure the shadows ofEarth. The mind belonged toHeaven, the body's shadow lies here.

Leopold Kronecker (1823-1891)y God created the natural number, and all the rest is the work of many Of what use is your (Lindemann's proof of transcendental of pi) beautiful investigation regarding

pi ? Why study such problems when irrational numbers do not exist ?

y Number theorists are like lotus-eaters - having tasted this food they can never give it up.Joseph Louis Lagrange (1736-1813)


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y I must meditate further on this (this refers to Parallel Postulate)y I regard as quite useless the reading of large treatises of pure analysis: too large a number of

methods pass at once before the eyes. It is in the works of applications that one must study

them; one judges their ability there and one apprises the manner of making use of them.

y As long as algebra and geometry have been separated, their progress have been slow and theiruses limited; but when these two sciences have been united, they have lent each mutual forces,

and have marched together towards perfection.

Johann Lambert (1728-1777)

y This hypothesis (Parallel hypothesis) would not destroy itself at all easily.y I should almost therefore put forward the proposal that the third hypothsis (angle sum of a

triangle less than two right angles) holds on the surface of an imaginary sphere.

y Proofs of the Euclidean [parallel] postulate can be developed to such an extent that apparently amere trifle remains. But a careful analysis shows that in this seeming trifle lies the crux of the

matter; usually it contains either the proposition that is being proved or a postulate equivalent

to it.y I am undecided wheter or not the Milky Way is but one of countless others all of which form an

entire system. Perhaps the light from these infinitely distant galaxies is so faint that we cannot

see them.

ierre-Simon Laplace (1749-1827)

y The invention of logarithms, by shortening the labors, double the life of the astronomer.y All the effects of nature are only mathematical results of a small number of immutable laws.y What we know is not much. What we do not know is immense.y Read Euler: he is our master in everything.y It is India that gave us the ingenious method of expressing all numbers by means of ten symbols,

each symbol receiving a value of position as well as an absolute value; a profound and important

idea which appears so simple to us now that we ignore its true merit. But its very simplicity and

the great ease which it has lent to computations put our arithmetic in the first rank ofuseful

inventions; and we shall appreciate the grandeur of the achievement the more when we

remember that it escaped the genius ofArchimedes and Apollonius, two of the greatest men

produced by antiquity.

y It is interesting thus to follow the intellectual truths of analysis in the phenomena of nature. Thiscorrespondence, of which the system of the world will offer us numerous examples, makes one

of the greatest charms attached to mathematicall speculations.

Andrien-Marie Legendre (1752-1833)

y These ... tables (values of trignometry functions), constructed by means of new techniquesbased principally on the calculus of differences, are one of the most beautiful monuments ever

erected to science.

y It is a matter for considerable regret that Fermat, who cultivated the theory of numbers with somuch success, did not leave us with the proofs of the theorems he discovered. In truth, Messrs

Euler and Lagrange, who have not distained this kind of research, have proved most of these


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theorems, and have even substituted extensive theories for the isolated propositions ofFermat.

But there are several proofs which have resisted their efforts.

Gottfried Wilhelm Leibniz (1646-1716)

y Taking mathematics from the beginning of the world to the time when Newton lived, what hehad done was much the better half.

y It is rare to find learned men who are clean, do not stink and have a sense of humour. (refers toCharles Louis De Secondat Montesquieu and to the Dutches of Orleans)

y Nothing is more important than to see the sources of invention which are, in my opinion moreinteresting than the inventions themselves.

y The pleasure we obtain from music comes from counting, but counting unconsciously. Music isnothing but unconscious arithmetic.

y He who understands Archimedes and Apollonius will admire less the achievements of theforemost men of later times.

y The art of discovering the causes of phenomena, or true hypothesis, is like the art ofdecyphering, in which an ingenious conjecture greatly shortens the road.

y The imaginary number is a fine and wonderful resource of the human spirit, almost anamphibian between being and not being.

y Miracles are not to be multiplied beyond necessity.

John Von Neumann (1903-1957)

y We must regard classical mathematics as a combinatorial game played with symbols.Sir Isaac Newton (1643-1727)

y If I have been able to see further, it was only because I stood on the shoulders of giants.y It is the glory of geometry that from so few principles, fetched from without, it is able to

accomplish so much.

y ...from the same principles, I now demonstrate the frame of the System of the World. (PrincipiaMathematica)

y I feign no hypotheses.y The description of right lines and circles, upon which geometry is founded, belongs to

mechanics. Geometry does not teach us to draw these lines, but requires them to be drawn.

y The latest authors, like the most ancient, strove to subordinate the phenomena of nature to thelaws of mathematics.

y God created everything by number, weight and measure.y I will not define time, space, place and motion, as being well known to all.y Truth is ever to be found in the simplicity, and not in the multiplicity and confusion of things.

Blaise Pascal (1623-1662)(3rd children of Etienne Pascal

(1588-1640))

y It is the heart which perceives God and not the reason.


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y Our nature consists in movement; absolute rest is death.y Everything that is written merely to please the author is worthless.y I cannot judge my work while I am doing it. I have to do as painters do, stand back and view it

from a distance, but not too great a distance. How great? Guess.

y Contradiction is not a sign of falsity, nor the lack of contradiction a sign of truth.y Nature is an infinite sphere of which the center is everywhere and the circumference nowhere.y It is not certain that everything is uncertain.y Always substitute mentally the definitions in place of the defined.y One must have altogether a wrong spirit to reason badly about principles so great that it is

almost impossible for them to escape.

y To speak freely of mathematics, I find it the highest exercise of the spirit; but at the same time Iknow that it is so useless that I make little distinction between a man who is only a

mathematician and a common artisan. Also, I call it the most beautiful profession in the world;

but it is only a profession;

y Humble thyself, impotent reason.y The excitement that a gambler feels when making a bet is equal to the amount he might win

times the probability of winning it.

y The more I see of men, the better I like my dog.y I have made this letter longer than usual, because I lack the time to make it short.

Plato (429-347 B. C. E), Plato's Academy

y Let no one ignorant of geometry enter here.y Whatever we Greeks receive we improve and perfect.y Geometry will draw the soul t oward truth and create the spirit of philosophy.y No nature except an extraordinary one could ever easily formulate a theory.y He who can properly define and divide is to be considered a god.y The ludicrous state of solid geometry made me pass over this branch.y He is unworthy of the name of man who is ignorant of the fact that the diagonal of a square is

incommensurable with its side.

y Mathematics is like draughts [checkers] in being suitable for the young, not too difficult,amusing, and without peril to the state.

y The knowledge of which geometry aims is the knowledge of the eternal.y I have hardly ever known a mathematician who was capable of reasoning.y ... arithmetic is a kind of knowledge in which the best natures should be trained, and which must

not be given up.

Henri Poincare (1854-1912)

y How can intuition deceive us at this point ?y Facts do not speak.y One geometry cannot be more true than another; it can only be more convenient.y It (logic) is no longer sterile, it begets contradictions.y Later mathematicians will regard set theory as a disease from which one has recovered.y Mathematicians are born, not made.y I entered an omnibus to go to some place or other. At that moment when I put my foot on the

step the idea came to me, without anything in my former thoughts seeming to have paved the


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way for it, that the transformations I had used to define the Fuchsian functions were identical

with non-Euclidean geometry.

y Mathematics is the art of giving the same name to different things. [As opposed to thequotation: Poetry is the art of giving different names to the same thing].

y Science is built up with facts, as a house is with stones. But a collection of facts is no more ascience than a heap of stones is a house.

y A scientist worthy of his name, about all a mathematician, experiences in his work the sameimpression as an artist; his pleasure is as great and of the same nature.

y Mathematicians do not study objects, but relations between objects. Thus, they are free toreplace some objects by others so long as the relations remain unchanged. Content to them is

irrelevant: they are interested in form only.

y Thought is only a flash between two long nights, but this flash is everything.y Mathematical discoveries, small or great are never born of spontaneous generation They always

presuppose a soil seeded with preliminary knowledge and well prepared by labour, both

conscious and subconscious.

y Zero is the number of objects that satisfy a condition that is never satisfied. But as never means"in no case", I do not see that any progress has been made.

Matthew Prior (1664-1721)

y Circles to square and cubes to double would give a man exercise trouble.Diadochus Proclus (410-485)

y Wherever there is number, there is beauty.y According to most accounts, geometry was first discovered among the Egyptians, taking its

origin from the measurement of areas. For they found it necessary by reason of the flooding of

the Nile, which wiped out everybody's proper boundaries. Nor is there anything surprising inthat the discovery both of this and of the other sciences should have had its origin in a practical

need, since everything which is in process of becoming progresses from the imperfect to the

perfect.

y The Pythagoreans considered all mathematical science to be divided into four parts: one halfthey marked off as concerned with quantity, the other half with magnitude; and each of these

they posited as twofold. A quantity can be considered in regard to its character by itself or in

relation to another quantity, magnitudes as either stationary or in motion. Arithmetic, then,

studies quantity as such, music the relations between quantities, geometry magnitude at rest,

spherics magnitude inherently moving.

y This, therefore, is Mathematics:She reminds yo

u

of the invisible forms of the sou

l;she gives life to her own discoveries;

she awakens the mind and purifies the intellect;

she brings to light our intrinsic ideas;

she abolishes oblivion and ignorance which are ours by birth ...

Pythagoras (572-497 B.C. E) and Pythagorean


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y All was numbers.y Number was the substance of all things.y Number rules the universe.y Number is the ruler of forms and ideas, and the cause of gods and demons.y Geometry is knowledge of the eternally existent.y There is geometry in the humming of the strings.

Bernard Riemann (1826-1866)

y If only I had the theorems! Then I should find the proofs easily enough.y Therefore, either the reality on which our space is based must form a discrete manifold or else

the reason for the metric relationships must be sought for, externally, in the binding forces

acting on it.

y What remains to be resovled is the question of knowing to what extent and up to what pointthese hypotheses are found to be confirmed by experience.

y It is well known that geometry presupposes not only the concept of space but also the firstfundamental notions for constructions in space as given in advance. It only gives nominaldefinitions for them, while the essential means of determining them appear in the form of

axioms. The relationship of these presumptions is left in the dark; one sees neither whether and

in how far their connection is necessary, nor a priori whether it is possible. From Euclid to

Legendre, to name the most renowned of modern writers on geometry, this darkness has been

lifted neither by the mathematicians nor the philosophers who have laboured upon it.

Joseph Alfred Serret (1819-1885)

y Algebra is, properly speaking, the Analysis of equations.

Socrates (469-399 B. C. E)

y The understanding of mathematics is necessary for a sound grasp of ethics.Alfred North Whitehead (1861-1947)

y Algebra reverses the relative importance of the factors in ordinary language.y Everything of importance has been said before by somebody who did not discover it.y Seek simplicity, and distrust it.y Fundamental progress has to do with the reinterpretation of basic ideas.y We think in generalities, but we live in details.y "Necessity is the mother of invention" is a silly proverb. "Necessity is the mother of futile

dodges" is much nearer the truth.

y Let us grant that the pusuit of mathematics is a divine madness of the human spirit.y Order is not sufficient. What is required, is something much more complex. It is order entering

upon novelty; so that the massiveness of order does not degenerate into mere repetition; and

so that the novelty is always reflected upon a background of system.


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y No Roman ever lost his life because he was absorbed in the contemplation of a mathematicaldiagram.

y Algebra is the intellectual instrument which has been create for rendering clear the quantitativeaspect of the world.

y The science of pure mathematics may claim to be the most original creation of the human spirit.Hermann Klaus Hugo Weyl (1885-1955)

y I would like to throttle the man who wrote this book.y God exists since mathematics is consistent, and the devil exists since its consistency cannot be

proved.

y If the game of mathematics is actually consist, then the formula of consistency cannot be provedwithin this game.

y In our time, the angel of topology and the devil of abstract algebra are fighting for everymathematical domain.

y Logic is the hygiene the mathematician practices to keep his ideas healthy and strong.y

Symmetry, as wide or narrow as you may define its meaning, is one idea by which man throughthe ages has tried to comprehend and create order, beauty, and perfection.

Xenophanes (570-475 B. C. E)

y The gods did not reveal all things to men at the start; but as time goes on, by searching, theydiscover more and more.

Ernst Zermelo (1871-1953)

y No mathematical error can be demonstrated in my [earlier] proof.

y ...self-evidence ... must not be confused with ... provability.

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