Some Math Quotes

• To think the thinkable -- that is the mathematician's aim. -- C. J. Keyser
• This mighty maze is not without a plan. -- Alexander Pope
• In most sciences one generation tears down what another has built and what one has established another undoes. In Mathematics alone each generation builds a new story to an old structure. -- Hermann Hankel
• I'v got it -- Jean Francois Champollion, as he unlocked the secrets to the Rossetta Stone and immediatly fainted.
• He is unworthy of the name of man who does not know that the diagonal of a square is incommensurable with its side. -- Plato
• Let no man ignorant of geometry enter here. -- Plato, inscription over the door to his academy.
• If I have seen farther than others, it is becuase I have stood on the shoulders of giants. -- Isaac Newton
• Geometry in every proposition speaks a language which experience never dares to utter; and indeed of which she but halfway comprehends the meaning. -- William Whewell.
• I have always regarded mathematics as an object of amusement rather than of ambition, and I can assure you that I enjoy the works of others much more than my own, with which I am always dissatisfied. -- Joseph-Louis Lagrange
• When we ask advice, we are usually looking for an accomplice. -- Joseph-Louis Lagrange
• Profound study of nature is the most fertile source of mathematical discoveries -- Joseph Fourier
• It matters little who first arrives at an idea, rather what is significant is how far that idea can go -- Sophie Germain
• Transcendental [numbers], They transcend the power of algebraic methods. -- Leonhard Euler
• No contradictions will arise as long as Finite Man does not mistake the infinite for something fixed, as long as he is not led by an acquired habit of mind to regard the infinite as something bounded. -- Carl Friedrich Gauss
• This winter I am giving two courses of lectures to three students, of whom one is only moderately prepared, the other less than moderately, and the third lacks both preparation and ability. Such are the burdens of a mathematical calling. -- Carl Friedrich Gauss
• Every assertion that is made is fully proved, and the assertions succeed one another in a perfectly just and logical order; there is nothing so far of which we can complain. But when we have finished the perusal, we soon begin to feel that our work is but begun, that we are still standing on the threshold of the temple, and that there is a secret which lies behind the veil and is yet concealed from us. -- Henry Smith on Gauss' works
• The analysts try in vain to conceal the fact that they do not deduce: they combine, they compose ... when they do arrive at the truth they stumble over it after groping their way along. --- Evariste Galois
• The Devine Spirit found a sublime outlet in that wonder of analysis -- Gottfried Wilhelm Von Leibnitz
• Life is only good for two things, doing mathematics and teaching it. -- Simeon-Denis Poisson
• The sole object of the science is the honor of the human spirit and that on this view a problem in the theory of numbers is worth as much as a problem of the solar system. -- Carl Jacobi
• Dirichlet alone, not I, nor Cauchy, nor Gauss knows what a completely rigorous proof is. Rather we learn it first from him. When Gauss says he has proved something it is clear; when Cauchy says it, one can wager as much pro as con; when Dirichlet says it, it is certain. -- Carl Jacobi
• The glory of science consists in its having no use. -- Carl Jacobi
• It is self-evident that any and all paths must be open to a researcher during the actual course of his [or her] investigations. -- Karl Weierstrass
• The infinite! No other question has ever moved so profoundly the spirit of man; no other idea has so fruitfully stimulated his intellect; yet no other concept stands in greater need of clarification that that of the inifinite. -- Karl Weierstrass
• A function f(x) has a limit L at x=x₀ if for any postive number &epsilon, there exists a &delta such that |f(x) - L| < &epsilon for all x in the deleted interval 0 < |x - x₀| < &delta . -- Karl Weierstrass
• The irrational number, logically defined, is an intellectual monster. -- unknown.
• [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
• Logic is barren, where mathematics is the most prolific of mothers. - unknown.
• Mathematical sciences have attracted special attention since great antiquity; they are attracting still more attention today because of their influence on industry and the arts. The agreement of theory and practice brings most beneficial results; and it is not exclusively the practical side which gains; science is advancing under its influence as it discovers new objects of study and new aspects to investigate in objects long familiar. In spite of the great advance of the mathematical sciences due to the efforts of outstanding geometers over the last three centuries, in practice many imperfections appear. New problems emerge and new methods are required -- Pafnuty Lvovich Chebyshev
• In mathematics our role is more that of servant than master. -- Charles Hermite
• I turn away with fear and horror from this lamentable sore of continuous functions without derivatives. -- Charles Hermite
• I shall risk nothing on an attempt to show the transcendence of &pi . If others undertake it, no one will be happier than I at their success, but believe me, my dear friend, this cannot fail to cost them effort. -- Charles Hermite
• God Himself made the integers - everything else is the work of man. -- Leopold Kronecker
• Definitions must contain the means of reaching a decision in a finite number off steps, and existence proofs must be conducted so that the quantity in question can be calculated with any degree of accuracy. -- Leopold Kronecker
• Of what use is your beautiful investigation regardeing &pi ? Why study such problems when irrational numbers do not exists? -- Leopold Kronecker
• For mathematics, even to the logical forms in which it moves, is entirely dependent on the concept of natural number. -- Hermann Weyl
• For what I have accomplished and what I have become, I have to to thank my industry much more, my indefatigable working, rather than any outstanding talent. -- Richard Dedekind
• All of mathematics is a tale about groups. -- Henri Poincare
• Later mathematicians will regard set theory as a disease from which one has recovered. -- Henri Poincare
• 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. -- Henri Poincare
• But all of my efforts served only to make me better acquainted with the difficulty, which in itself was something. -- Henri Poincare
• The intuition, by which discoveries are made, is a direct communion, without possible intermediaries, with the spirit and the truth -- A comment from Henri Poincare's nephew about his uncle's beliefs.
• Logic sometimes makes monsters. For half a century we have seen a mass of bizarre functions which appear to be forced to resemble as little as possible honest functions which serve some purpose. -- Henri Poincare
• No more fiction for us: we calculate; but that we may calculate, we had to make fiction first. -- Nietzsche
• Without the perpetual counterfeiting of the universe by number, man could not continue to live -- Nietzsche
• The divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever. -- Niels Henrik Abel
• I see it but I don't believe it! -- Georg Cantor
• The essence of mathematics is its freedom. -- Georg Cantor
• Astronomy and mathematics were in their time regarded as inseparable. With the succeeding generation, however, the tendency to specializeation manifests itself. The developing science departs at the same time more and more from its original scope and purpose and threatens to sacrifice its earlier unity and split into diverse branches. -- Felix Klein
• That is not mathematics. That is theology -- Paul Gordan's comment after David Hilbert proved a result on invariants.
• Oh, it is very useful indeed; once can write many theses about it. -- Paul Gordan
• We ought not to believe those who today, adopting a philosophical air and with a tone of superiority, prophesy the decline of culter and are content with the unknowable in a self-satisfied way. For us there is no unknowable, and in my opinion there is also non whatsoever for the natural sciences. In place of this foolish unknowable, let our watchword on the contrary be: we must know - we shall know. -- David Hilbert
• The impact of a scientist on his epoch is not directly proportional to the scientific weight of his research. -- Hermann Weyl
• That teacher teaches best who teaches least. -- Robert Lee Moore
• 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
• 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
• Nature is written in mathematical language -- Galileo Galilei
• Any argument where one supposes an arbitrary choice to be made an uncountably infinite number of times ...[is] outside the domain of mathematics. -- Emile Borel
• Can the existence of a mathematical entity be proved without defining it? -- Jacques Hadamard
• I assert that, in any particular natural science, one encounters genuine scientific substance only to the extent that mathematics is present. -- Immanuel Kant
• We do not master a scientific theory until we have shelled and completely prised free its mathematical kernel. -- David Hilbert
• Taking the law of the excluded middle from mathematicians is the same as prohibiting the astronomer his telescope or the boxer the user of his fists. --
• More than any other science, mathematics develops through a sequence of successive abstractions. A desire to avoid mistakes forces mathematicians to find and isolate the essence of the problems and entities considered. -- Elie Cartan
• In mathematical activity, as in any other type of human activity, one should find a balance of values: there is not doubt that it is important to think correctly, but it is even more important to formulate the right problems. -- Elie Cartan
• I have never done anything `useful'. 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.... The case for my life, then, or for that of anyone else who has been a mathematician in the same sense in which I have been one, is this: that I have added something to knowledge, and helped others to add more; and that these somethings have a value which differs in degree only, and not in kind, from that of the creations of the great mathematicians, or of the other artists, great or small, who have left some kind of memorial behind them. -- Godfrey Harold Hardy
• 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. -- Godfrey Harold Hardy
• He always seemed to feel that his leisure was scanty, and that he dare not risk squandering any part of it on a difficult problem that might lead nowhere; and of course when a fine mathematician begins to feel like that there is always a serious danger that he may not do justice to his [or her] powers. -- Godfrey Harold Hardy commenting on George Polya
• Here's to pure mathematics! May it never have any use -- Godfrey Harold Hardy
• It is curious that so often mathematical genius goes hand in hand with freedom of judgement and noble character. -- anonymous
• People first have to make up their minds and then find their reasons. -- Solomon Lefschetz
• An equation has no meaning for me unless it expresses a thought of God. -- Srinivasa Iyengar Ramanujan
• I tell you that studying humanities in high school is more important than mathematics - mathematics is too sharp an instrument, no good for kids. -- Stefan Banach
• Am I really a good mathematician? -- Norbert Wiener
• You should love the work you do, and you should love it disinterestedly, and do it because you love it, and not because you expect personal success, or any sort of profit for yourself from it. -- Pavel Sergeevitch Aleksandrov
• Geometry is the real life. -- Oscar Zariski
• Well, if it's just turning the crank, it's algebra, but if it's got an idea, it's topology. -- Oscar Zariski
• We aristocrats do not need proofs, proofs are for you plebeians. -- Federigo Enrique
• How then shall mathematical concepts be judged? They shall not be judged. Mathematics is the supreme arbiter. From its decisions there is no appeal. We cannot change the rules of the game, we cannot ascertain whether the game is fair. We can only study the player at his game; not, however, with the detached attitude of a bystander, for we are watching our own minds at play. -- Tobias Dantzig
• Are not most professional mathematicians spared all trouble incident to income? -- Tobias Dantzig
• The practical man demands an appearance of reality at least. Always dealing in the concrete, he regards mathematical terms not as symbols or thought but as images of reality. A system acceptable to the mathematician because of its inner consistency may appear to the practical man to be full of contradictions because of the incomplete manner in which it represents reality. -- Tobias Dantzig
• The importance of infinite processes for the practical exigencies of technical life can hardly be overemphasized. Practically all applications of arithmetic to geometry, mechanics, physics and even statistics involve these processes directly and indirectly. -- Tobias Dantzig
• the infinite process succeeded where the rational number had failed. -- Tobias Dantzig
• Vigorous writing is concise. -William Strunk
• What is beautiful and definite and the object of knowledge is by nature prior to the indefinite and the incomprehensible and the ugly. -- Nicomachus
• All things that have been arranged by nature according to a workmanlike plan appear, both individually and as a whole, as singled out and set in order by Foreknowledge and reason, which created all according to Number, conceivable to mind only and therefore wholly immaterial; yet real, indeed, the really real, the eternal. -- Nicomacus
• All thinking which can be known must have number; for it is not possible that without number anything can be eiher conceived or known. -- Philolaus
• But what has been said once, can always be repeated. -- Zeno of Elea
• One cannot escape the feeling that these mathematical formulae have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we got more of them than was originally put into them. -- Heinrich Hertz
• We must learn a new modesty. We have stormed the heavens, but succeeded only in building fog upon fog, a mist which will not support anybody who earnestly desires to stand upon it. What is valid seems so insignificant that it may be seriously doubted whether anlaysis is at all possible. -- Hermann Weyl
• Mathematics is the science which draws necessary conclusions. -- Benjamin Peirce
• Gentlemen, this is surely true, it is absolutely pradaxical, we can't understand it, and we haven't the slightest idea what the equation means, but we may be sure that it means something very important. -- Benjamin Peirce commenting on Euler's equation
• I believe in extreme points. -- George Nemhauser
• We have found a strange footprint on the shores of the unknown. We have devised profound theories, one after another, to account for its origin. At last, we have succeeded in reconstructing the creature that made the footprint. An lo! it is our own. -- A. S. Eddington
• The solution of the difficulties which formerly surrounded the mathematical infinite is probably the greatest achievement of which our age has to boast -- Bertrand Russell
• Is the set of all sets which are not members of themselves a member of itself? -- Bertrand Russell
• Thank God that number theory is unsullied by applications. -- Leonard Eugene Dickson
• Trivial -- John von Neumann's comment on John Nash's dissertation, which later won Nash the Nobel Prize.
• However gemlike mathematical truths may be, research is but a human endeavor. -- David Burton
• The fact that a mathematical result may or may not apply directly to a problem in another discipline does not affect its intrinsic value. If the genisis of the problem does not originate in mathematics, who cares, for the solution does. -- Allen Holder