Quotes of All Topics . Occasions . Authors
As for my memory, I have a particularly good one. I never keep any record of my investigations or experiments. My memory files all these things away conveniently and reliably. I should say, though, that I didn’t cumber it up with a lot of useless matter.
Let me also remind you that zero, like all of mathematics, is fictional and an idealization. It is impossible to reach absolute zero temperature or to get perfect vacuum. Luckily, mathematics is a fairyland where ideal and fictional objects are possible.
The fear of infinity is a form of myopia that destroys the possibility of seeing the actual infinite, even though it in its highest form has created and sustains us, and in its secondary transfinite forms occurs all around us and even inhabits our minds.
Those who become incoculated with the virus of race hatred are more unfortunate than the victim of it. Race hatred is the most malignant poison that can afflict the mind. It freezes up the fount of inspiration and chills the higher faculties of the soul.
The divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever. By using them, one may draw any conclusion he pleases and that is why these series have produced so many fallacies and so many paradoxes.
The problems of financing the universities and their intellectual freedom, threatened by political and bureaucratic interference, are problems which are invariant under the ism transformations: socialism, communism, capitalism, or any other ism or ology.
While I was the mayor of Bogotá, I received occasional death threats. Therefore, I had to use a bullet-proof vest. I made a hole right where my heart is. The hole was in the shape of a heart. I believe this kind of gesture, gave me indeed more protection.
When malice has reason on its side, it looks forth bravely, and displays that reason in all its luster. When austerity and self-denial have not realized true happiness, and the soul returns to the dictates of nature, the reaction is fearfully extravagant.
These Gentlemen must be told, that they take too much upon themselves when they pretend to appoint how far and no farther Men shall go in their Searches, and to set bounds to other Mens Industry; as if they knew the Marks that God has placed to Knowledge.
As soon as somebody demonstrates the art of flying, settlers from our species of man will not be lacking on the moon and Jupiter... Given ships or sails adapted to the breezes of heaven, there will be those who will not shrink from even that vast expanse.
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.
Man is to himself the most wonderful object in nature; for he cannot conceive what the body is, still less what the mind is, and least of all how a body should be united to a mind. This is the consummation of his difficulties, and yet it is his very being.
Our senses will not admit anything extreme. Too much noise confuses us, too much light dazzles us, too great distance or nearness prevents vision, too great prolixity or brevity weakens an argument, too much pleasure gives pain, too much accordance annoys.
The validity of mathematical propositions is independent of the actual world-the world of existing subject-matters-is logically prior to it, and would remain unaffected were it to vanish from being. Mathematical propositions, if true, are eternal verities.
The right to live does not connote the right of each man to reproduce his kind ... As we lessen the stringency of natural selection, and more and more of the weaklings and the unfit survive, we must increase the standard, mental and physical, of parentage.
Art heightens the sense of humanity. It gives an elation to feeling which is supernatural...A million sunsets will not spur us on towards civilization. It requires Art to evoke into consciousness the finite perfections which lie ready for human achievement.
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.
Human beings do not know their place and purpose. They have fallen from their true place, and lost their true purpose. They search everywhere for their place and purpose, with great anxiety. But they cannot find them because they are surrounded by darkness.
From whence comes it that a cripple in body does not irritate us, and that a crippled mind enrages us? It is because a cripple sees that we go right, and a distorted mind says that it is we who go astray. But for that we should have more pity and less rage.
We must wholeheartedly believe in free will. If free will is a reality, we shall have made the correct choice. If it is not, we shall still not have made an incorrect choice, becauee we shall not have made any choice at all, not having a free will to do so.
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.
To unfold the secret laws and relations of those high faculties of thought by which all beyond the merely perceptive knowledge of the world and of ourselves is attained or matured, is a object which does not stand in need of commendation to a rational mind.
In whatever manner God created the world, it would always have been regular and in a certain general order. God, however, has chosen the most perfect, that is to say, the one which is at the same time the simplest in hypothesis and the richest in phenomena.
There is no study in the world which brings into more harmonious action all the faculties of the mind than [mathematics], ... or, like this, seems to raise them, by successive steps of initiation, to higher and higher states of conscious intellectual being.
A faith which we follow upon orders imposed from outside is no faith, and a community which puts its dependence upon such a pseudo-faith is ultimately bound to ruin itself because of the paralysis which the lack of a healthy growing science imposes upon it.
Corporate America needs to get its act together to see that the education system is changed so it produces what it needs. The educational system that teaches kids to be passive recipients of knowledge worked when most workers were sitting in assembly lines.
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.
If I steal money from any person, there may be no harm done from the mere transfer of possession; he may not feel the loss, or it may prevent him from using the money badly. But I cannot help doing this great wrong towards Man, that I make myself dishonest.
It is certain that the soul is either mortal or immortal. The decision of this question must make a total difference in the principles of morals. Yet philosophers have arranged their moral system entirely independent of this. What an extraordinary blindness!
Thus so wretched is man that he would weary even without any cause for weariness... and so frivolous is he that, though full of a thousand reasons for weariness, the least thing, such as playing billiards or hitting a ball, is sufficient enough to amuse him.
Now since in so many Things they... agree, what can be more probable than that in others they agree too; and that the other Planets are as beautiful and as well stock'd with Inhabitants as the Earth? Or what shadow of Reason can there be why they should not?
If you're setting a game during the Cuban Missile Crisis, look through a library. find out what people were wearing, what other issues were in the news, how houses were furnished, what cars were being driven. Especially include things which now seem foreign.
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.
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.
It is impossible to overstate the imporance of problems in mathematics. It is by means of problems that mathematics develops and actually lifts itself by its own bootstraps... Every new discovery in mathematics, results from an attempt to solve some problem.
Cold words freeze people, and hot words scorch them, and bitter words make them bitter, and wrathful words make them wrathful. Kind words also produce their own image on men's souls; and a beautiful image it is. They smooth, and quiet, and comfort the hearer.
God is, or He is not." But to which side shall we incline? Reason can decide nothing here. There is an infinite chaos which separated us. A game is being played at the extremity of this infinite distance where heads or tails will turn up. What will you wager?
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.
When it comes to scientific matters the ready talkers simply run riot. There are a lot of pseudo-scientists who with a little technical jargon to spatter through their talk are always getting in the limelight... The less they know the surer they are about it.
The potential infinite means nothing other than an undetermined, variable quantity, always remaining finite, which has to assume values that either become smaller than any finite limit no matter how small, or greater than any finite limit no matter how great.
Euclid 's manner of exposition, progressing relentlessly from the data to the unknown and from the hypothesis to the conclusion, is perfect for checking the argument in detail but far from being perfect for making understandable the main line of the argument.
I saw the first one [video with Hans Rosling ] when he did - I think it was his first one - in 2006, a TED Talk. And for the first time in my life, I thought here's someone who can take statistics that most people regard as dull and boring and bring it alive.
It may be said "In research, if you know what you are doing, then you shouldn't be doing it." In a sense, if the answer turns out to be exactly what you expected, then you have learned nothing new, although you may have had your confidence increased somewhat.
Thanks to my memory, which enabled me to quote Latin and to discuss Greek and Roman civilization, it became obvious to some of my colleagues in other fields that I was interested in things outside mathematics. This lead quickly to very pleasant relationships.
Experience has shown repeatedly that a mathematical theory with a rich internal structure generally turns out to have significant implications for the understanding of the real world, often in ways no one could have envisioned before the theory was developed.
The teleology of the Universe is directed to the production of Beauty... The type of Truth required for the final stretch of Beauty is a discovery and not a recapitulation... Apart from Beauty, Truth is neither good, nor bad... Truth matters because of beauty.
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.
The mathematician is in much more direct contact with reality. ... [Whereas] the physicist's reality, whatever it may be, has few or none of the attributes which common sense ascribes instinctively to reality. A chair may be a collection of whirling electrons.
For a long time the objects that mathematicians dealt with were mostly ill-defined; one believed one knew them, but one represented them with the senses and imagination; but one had but a rough picture and not a precise idea on which reasoning could take hold.
In defiance of all the tortue, of all the might, of all the malice of the world, the liberal man will ever be rich; for God's providence is his estate, God's wisdom and power are his defence, God's love and favor are his reward, and God's word is his security.