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

- 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.
- There is nothing strange in the circle being the origin of any and every marvel.
- 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.
- To Thales the primary question was not what do we know, but how do we know it.
- 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.
- It is not once nor twice but times without number that the same ideas make their appearance in the world.
- But Nature flies from the infinite, for the infinite is unending or imperfect, and Nature ever seeks an end.
- We cannot ... prove geometrical truths by arithmetic.
- The chief forms of beauty are order and symmetry and definiteness, which the mathematical sciences demonstrate in a special degree.
- The continuum is that which is divisible into indivisibles that are infinitely divisible. Physics.

- In mathematics I can report no deficience, except it be that men do not sufficiently understand the excellent use of the Pure Mathematics.
- Mathematics is the door and key to the sciences.
- 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.
- There are four great sciences ... Of these sciences the gate and key is mathematics, which the saints discovered at the beginning of the world.
- ... mathematics is absolutely necessary and useful to the other sciences.

- One can always reason with reason.

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

- My special pleasure in mathematics rested particularly on its purely speculative part.
- 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.

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

- The essence of mathematics is its freedom.
- I see it, but I don't believe it.
- 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.
- In mathematics the art of proposing a question must be held of higher value than solving it.
- every transfinite consistent multiplicity, that is, every transfinite set, must have a definite aleph as its cardinal number.

- I think, therefore I am.
- Perfect numbers like perfect men are very rare.
- With me everything turns into mathematics.
- It is not enough to have a good mind. The main thing is to use it well.
- ... the two operations of our understanding, intuition and deduction, on wh ich alone we have said we must rely in the acquisition of knowledge.
- In order to seek truth it is necessary once in the course of our life to doubt as far as possible all things.
- Give me extension and motion and I will construct the universe.
- There have been only Mathematicians who were able to find some proofs, that is to say some sure and certain reasons.
- Take what you need; act as you must, and you will obtain that for which you wish!
- Only having one truth about each object, whoever finds it knows as much as can known about it.

- Found, but not proven.
- 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.
- I would rather discover one scientific fact than become King of Persia.
- Everything existing in the Universe is the fruit of chance and necessity.
- Nothing exists except atoms and empty space; everything else is opinion.

- So far as the theories of mathematics are about reality, they are not certain; so far as they are certain, they are not about reality.
- I don't believe in mathematics.
- God does not care about our mathematical difficulties. He integrates empirically.
- Nature to him (Newton) was an open book, whose letters he could read without effort.
- Since the mathematicians have invaded the theory of relativity, I do not understand it myself anymore.
- Do not worry about your difficulties in mathematics, I assure you that mine are greater.

- There is no royal road to geometry.
- 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."

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

- Mighty is geometry; joined with art, resistless.

- To divide a cube into two other cubes, a fourth power or in general any power whatever into two 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.
- And perhaps, posterity will thank me for having shown it that the ancients did not know everything.

- Arithmetic has began to totter.
- Every good mathematician is at least half a philosopher, and every good philosopher is at least half a mathematician.
- What are numbers? What is the nature of arithmetical truth?

- Thou, nature, art my goddess; to thy laws My Services are bound.
- I am ever more convinced that the necessity of our geometry cannot be proved -- at least not by human reason for human reason.
- One sees by that, what should be understood by a class composed from two or more class.
- Mathematics is the queen of the sciences and number theory is the queen of mathematics.
- The total number of Dirichlet's publications is not large: jewels are not weighed on a grocery scale.
- 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.
- God does arithmetic.
- I have had my results for a long time: but I do not yet know how I am to arrive at them.
- Mathematics is concerned only with the enumeration and comparison of relations.

- Nothing new had been done in Logic since Aristotle!
- Either mathematics is too big for the human mind or the human mind is more than a machine.
- I don't believe in natural science.
- 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.
- ...a consistency proof for [any] system ... can be carried out only by means of modes of inference that are not formalized in the system ... itself.

- This (axiomatic math) is no longer mathematics, it is theology.

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

- Can the existence of a mathematical entity be proved without definiing it ?
- The shortest path between two truths in the real domain passes through the complex domain.
- Practical application is found by not looking for it, and one can say that the whole progress of civilization rests on that principle.
- To parents who despair because their children are unable to master the first problems in arithmetic I can dedicate my examples. For, in arithmetic, until the seventh grade I was last or nearly last.
- Logic merely sanctions the conquests of the intuition.

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

- Turn away with fear and horror from this lamentable plague of continuous functions that do not have a derivative.
- 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 stuty them, as do the physicists, the chemists and the zoologists.
- 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.
- We are servants rather than masters in mathematics.
- 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.

- We must know, we will know.
- Let us consider three distinct systems of things. (refer to Chair, table, beer-mug as point, line and plane for axioms)
- Mathematics is a game played according to certain rules with meaningless marks on paper.
- No one will expel us from this paradise Cantor has created for us.
- The finest product (Cantor's work on set theory) of mathematical genius and one of the supreme achievments of purly intellectual human activity.
- I have tried to avoid long numerical computations, thereby following Riemann's postulate that proofs should be given through ideas and not voluminous computations.
- The art of doing mathematics consists in finding that special case which contains all the germs of generality.
- Mathematics knows no races or geographic boundaries; for mathematics, the cultural world is one country.
- The infinite! No other question has ever moved so profoundly the spirit of man.
- The faculty is not a pool changing room. [On the proposed appointment of the first woman professor.]
- If I were to awaken after having slept for a thousand years, my first question would be: Has the Riemann hypothesis been proven?

- It is a pity that you (Gauss) did not publish this result, since you have published so many poorer papers.
- It will not be possible to elucidate it (refers to Grammer rule for degenerate coefficent matrix) briefly.
- God ever arithmetizes.
- One should always generalize.
- The real end of science is the honour of the human mind.
- Mathematics is the science of what is clear by itself.
- The God that reigns in Olympus is Number Eternal.

- When you can measure what you are talking about and express it in numbers, you know something about it.

- However, before we come to [special] creation, which puts an end to all discussion: I think we should try everything else.
- 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.
- Truth is the daughter of time, and I feel no shame in being her midwife.
- ...I am stealing the golden vessels of the Egyptians to build a tabernacle to my God from them, far far away from the boundaries of Egypt. 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.
- I used to measure the Heavens, now I measure the shadows of Earth. The mind belonged to Heaven, the body's shadow lies here.

- God created the natural number, and all the rest is the work of man
- 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 ?
- Number theorists are like lotus-eaters - having tasted this food they can never give it up.

- I must meditate further on this (this refers to Parallel Postulate)
- 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.
- As long as algebra and geometry have been separated, their progress have been slow and their uses limited; but when these two sciences have been united, they have lent each mutual forces, and have marched together towards perfection.

- This hypothesis (Parallel hypothesis) would not destroy itself at all easily.
- 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.
- Proofs of the Euclidean [parallel] postulate can be developed to such an extent that apparently a mere 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.
- 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.

- The invention of logarithms, by shortening the labors, double the life of the astronomer.
- All the effects of nature are only mathematical results of a small number of immutable laws.
- What we know is not much. What we do not know is immense.
- Read Euler: he is our master in everything.
- 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 of useful inventions; and we shall appreciate the grandeur of the achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity.
- 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 mathematicall speculations.

- These ... tables (values of trignometry functions), constructed by means of new techniques based principally on the calculus of differences, are one of the most beautiful monuments ever erected to science.
- It is a matter for considerable regret that Fermat, who cultivated the theory of numbers with so much 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 theorems, and have even substituted extensive theories for the isolated propositions of Fermat. But there are several proofs which have resisted their efforts.

- Taking mathematics from the beginning of the world to the time when Newton lived, what he had done was much the better half.
- It is rare to find learned men who are clean, do not stink and have a sense of humour. (refers to Charles Louis De Secondat Montesquieu and to the Dutches of Orleans)
- Nothing is more important than to see the sources of invention which are, in my opinion more interesting than the inventions themselves.
- The pleasure we obtain from music comes from counting, but counting unconsciously. Music is nothing but unconscious arithmetic.
- He who understands Archimedes and Apollonius will admire less the achievements of the foremost men of later times.
- The art of discovering the causes of phenomena, or true hypothesis, is like the art of decyphering, in which an ingenious conjecture greatly shortens the road.
- The imaginary number is a fine and wonderful resource of the human spirit, almost an amphibian between being and not being.
- Miracles are not to be multiplied beyond necessity.

- We must regard classical mathematics as a combinatorial game played with symbols.

- If I have been able to see further, it was only because I stood on the shoulders of giants.
- It is the glory of geometry that from so few principles, fetched from without, it is able to accomplish so much.
- ...from the same principles, I now demonstrate the frame of the System of the World. (Principia Mathematica)
- I feign no hypotheses.
- 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.
- The latest authors, like the most ancient, strove to subordinate the phenomena of nature to the laws of mathematics.
- God created everything by number, weight and measure.
- I will not define time, space, place and motion, as being well known to all.
- Truth is ever to be found in the simplicity, and not in the multiplicity and confusion of things.

- It is the heart which perceives God and not the reason.
- Our nature consists in movement; absolute rest is death.
- Everything that is written merely to please the author is worthless.
- 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.
- Contradiction is not a sign of falsity, nor the lack of contradiction a sign of truth.
- Nature is an infinite sphere of which the center is everywhere and the circumference nowhere.
- It is not certain that everything is uncertain.
- Always substitute mentally the definitions in place of the defined.
- One must have altogether a wrong spirit to reason badly about principles so great that it is almost impossible for them to escape.
- To speak freely of mathematics, I find it the highest exercise of the spirit; but at the same time I know 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;
- Humble thyself, impotent reason.
- The excitement that a gambler feels when making a bet is equal to the amount he might win times the probability of winning it.
- The more I see of men, the better I like my dog.
- I have made this letter longer than usual, because I lack the time to make it short.

- Let no one ignorant of geometry enter here.
- Whatever we Greeks receive we improve and perfect.
- Geometry will draw the soul t oward truth and create the spirit of philosophy.
- No nature except an extraordinary one could ever easily formulate a theory.
- He who can properly define and divide is to be considered a god.
- The ludicrous state of solid geometry made me pass over this branch.
- 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.
- Mathematics is like draughts [checkers] in being suitable for the young, not too difficult, amusing, and without peril to the state.
- The knowledge of which geometry aims is the knowledge of the eternal.
- I have hardly ever known a mathematician who was capable of reasoning.
- ... arithmetic is a kind of knowledge in which the best natures should be trained, and which must not be given up.

- How can intuition deceive us at this point ?
- Facts do not speak.
- One geometry cannot be more true than another; it can only be more convenient.
- It (logic) is no longer sterile, it begets contradictions.
- Later mathematicians will regard set theory as a disease from which one has recovered.
- Mathematicians are born, not made.
- 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 way for it, that the transformations I had used to define the Fuchsian functions were identical with non-Euclidean geometry.
- Mathematics is the art of giving the same name to different things. [As opposed to the quotation: Poetry is the art of giving different names to the same thing].
- Science is built up with facts, as a house is with stones. But a collection of facts is no more a science than a heap of stones is a house.
- A scientist worthy of his name, about all a mathematician, experiences in his work the same impression as an artist; his pleasure is as great and of the same nature.
- Mathematicians do not study objects, but relations between objects. Thus, they are free to replace some objects by others so long as the relations remain unchanged. Content to them is irrelevant: they are interested in form only.
- Thought is only a flash between two long nights, but this flash is everything.
- 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.
- 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.

- Circles to square and cubes to double would give a man exercise trouble.

- Wherever there is number, there is beauty.
- 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 in that 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.
- The Pythagoreans considered all mathematical science to be divided into four parts: one half they 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.
- This, therefore, is Mathematics:

She reminds you of the invisible forms of the soul;

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 ...

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

- If only I had the theorems! Then I should find the proofs easily enough.
- 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.
- What remains to be resovled is the question of knowing to what extent and up to what point these hypotheses are found to be confirmed by experience.
- It is well known that geometry presupposes not only the concept of space but also the first fundamental notions for constructions in space as given in advance. It only gives nominal definitions for them, while the essential means of determining them appear in the form of axioms. The relationship of these presumptions is left in the dark; one sees neither whether and in how far their connection is necessary, nor a priori whether it is possible. From Euclid to Legendre, to name the most renowned of modern writers on geometry, this darkness has been lifted neither by the mathematicians nor the philosophers who have laboured upon it.

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

- The understanding of mathematics is necessary for a sound grasp of ethics.

- Algebra reverses the relative importance of the factors in ordinary language.
- Everything of importance has been said before by somebody who did not discover it.
- Seek simplicity, and distrust it.
- Fundamental progress has to do with the reinterpretation of basic ideas.
- We think in generalities, but we live in details.
- "Necessity is the mother of invention" is a silly proverb. "Necessity is the mother of futile dodges" is much nearer the truth.
- Let us grant that the pusuit of mathematics is a divine madness of the human spirit.
- 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.
- No Roman ever lost his life because he was absorbed in the contemplation of a mathematical diagram.
- Algebra is the intellectual instrument which has been create for rendering clear the quantitative aspect of the world.
- The science of pure mathematics may claim to be the most original creation of the human spirit.

- I would like to throttle the man who wrote this book.
- God exists since mathematics is consistent, and the devil exists since its consistency cannot be proved.
- If the game of mathematics is actually consist, then the formula of consistency cannot be proved within this game.
- In our time, the angel of topology and the devil of abstract algebra are fighting for every mathematical domain.
- Logic is the hygiene the mathematician practices to keep his ideas healthy and strong.
- Symmetry, as wide or narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty, and perfection.

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

- No mathematical error can be demonstrated in my [earlier] proof.
- ...self-evidence ... must not be confused with ... provability.

wli@math.okstate.edu