Quotes of All Topics . Occasions . Authors
Art forms that appeal to [leftists] tend to focus on ... defeat and despair ... as if there were no hope of accomplishing anything through rational calculation.
In every age of well-marked transition, there is the pattern of habitual dumb practice and emotion which is passing and there is oncoming a new complex of habit.
I didn't feel comfortable at first with pure mathematics, or as a professor of pure mathematics. I wanted to do a little bit of everything and explore the world.
When everyone is moving towards depravity, no one seems to be moving, but if someone stops he shows up the others who are rushing on, by acting as a fixed point.
When we wish to correct with advantage, and to show another that he errs, we must notice from what side he views the matter, for on that side it is usually true.
Notwithstanding the sight of all our miseries, which press upon us and take us by the throat, we have an instinct which we cannot repress, and which lifts us up.
Had Poincaré been as strong in practical science as he was in theoretical he might have made a fourth with the incomparable three, Archimedes, Newton, and Gauss.
When a truth is necessary, the reason for it can be found by analysis, that is, by resolving it into simpler ideas and truths until the primary ones are reached.
Indecision is debilitating; it feeds upon itself; it is, one might almost say, habit-forming. Not only that, but it is contagious; it transmits itself to others.
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].
For me, I guess the main motivation is the satisfaction of finally understanding some tricky mathematical concept or phenomenon and then explaining it to others.
The idea behind digital computers may be explained by saying that these machines are intended to carry out any operations which could be done by a human computer.
You think the world is what it looks like in fine weather at noon day; I think it is what it seems like in the early morning when one first wakes from deep sleep.
There is a tradition of opposition between adherents of induction and of deduction. In my view it would be just as sensible for the two ends of a worm to quarrel.
But you can catch yourself entertaining habitually certain ideas and setting others aside; and that, I think, is where our personal destinies are largely decided.
I tried to fit it in with some previous broad conceptual understanding of some part of mathematics that would clarify the particular problem I was thinking about.
Mathematics and art are quite different. We could not publish so many papers that used, repeatedly, the same idea and still command the respect of our colleagues.
If we must not act save on a certainty, we ought not to act on religion, for it is not certain. But how many things we do on an uncertainty, sea voyages, battles!
And is it not obvious that, just as it is a crime to disturb the peace when truth reigns, it is also a crime to remain at peace when the truth is being destroyed?
Too much pleasure disagrees with us. Too many concords are annoying in music; too many benefits irritate us; we wish to have the wherewithal to overpay our debts.
There are few circumstances which so strongly distinguish the philosopher, as the calmness with which he can reply to criticisms he may think undeservedly severe.
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.
It is a well-known experience that the only truly enjoyable and profitable way of studying mathematics is the method of "filling in details" by one's own efforts.
Analogy pervades all our thinking, our everyday speech and our trivial conclusions as well as artistic ways of expression and the highest scientific achievements.
For a fortnight nobody at all emailed me, or posted a follow-up. Doesn't anyone care, I thought? It turned out my newsreader was broken, and hadn't posted at all.
As Hamlet tells his friend, ``There are more things in heaven and earth, Horatio, than are dreamt of in your philosophy.'' Well then, we must try harder to dream!
For each successive class of phenomena, a new calculus or a new geometry, as the case might be, which might prove not wholly inadequate to the subtlety of nature.
A good problem should be more than a mere exercise; it should be challenging and not too easily solved by the student, and it should require some "dreaming" time.
I grew up to be indifferent to the distinction between literature and science, which in my teens were simply two languages for experience that I learned together.
I had the intention of becoming a theologian...but now I see how God is, by my endeavors, also glorified in astronomy, for 'the heavens declare the glory of God.'
I never saw my grandfather because he had died before I was born, but I have good memories of my grandmother and of how she could play the piano at the old house.
Racial and denominational schools impart to the membership of their communities something which the general educational institution is wholly unable to inculcate.
For a traveler going from any place toward the north, that pole of the daily rotation gradually climbs higher, while the opposite pole drops down an equal amount.
Mathematics is a field which has often been compared with chess, but differs from the latter in that it is only one's best moments that count and not one's worst.
Doing research in mathematics is frustrating and if being frustrated is something you cannot get used to, then mathematics may not be an ideal occupation for you.
There are two ways to do great mathematics. The first is to be smarter than everybody else. The second way is to be stupider than everybody else - but persistent.
For many years I was the youngest among my mathematical friends. It makes me melancholy to realize that I now have become the oldest in most groups of scientists.
Because we don't see the evil destroyed now and thus experience the suffering that evil inevitably inflicts, we are tempted to doubt God's existence and goodness.
Do you know why people like violence? It is because it feels good. Humans find violence deeply satisfying. But remove the satisfaction, and the act becomes hollow.
Just as all things speak about God to those that know Him, and reveal Him to those that love Him, they also hide Him from all those that neither seek nor know Him.
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 fact is there are few more popular subjects than mathematics. Most people have some appreciation of mathematics, just as most people can enjoy a pleasant tune.
Mathematics is the cheapest science. Unlike physics or chemistry, it does not require any expensive equipment. All one needs for mathematics is a pencil and paper.
Man masters nature not by force but by understanding. This is why science has succeeded where magic failed: because it has looked for no spell to cast over nature.
Knowledge is not a loose-leaf notebook of facts. Above all, it is a responsibility for the integrity of what we are, primarily of what we are as ethical creatures.
The most powerful drive in the ascent of man is his pleasure in his own skill. He loves to do what he does well. And having done it well, he loves to do it better.
Beyond all our actions stands the larger shadow: How are we to choose between what we have been taught to think right and something else which manifestly succeeds?
The object of pure physics is the unfolding of the laws of the intelligible world; the object of pure mathematics that of unfolding the laws of human intelligence.
One basic formula for understanding the Community is this: 'Take five broken empires, add the sixth one later, and make one big neo-colonial empire out of it all.'
I also ask you my friends not to condemn me entirely to the mill of mathematical calculations, and allow me time for philosophical speculations, my only pleasures.