Quotes of All Topics . Occasions . Authors
... mathematics is very much like poetry ... what makes a good poem -- a great poem -- is that there is a large amount of thought expressed in very few words. In this sense formulas like or are poems.
There is an old maxim that says that two empires that are too large will collapse. The analog in set theory is that two different theories that are too powerful must necessarily contradict each other.
There's a tendency to make jazzy educational software that's very uniform and therefore just like school. I'd like to see a company develop software for rebellious kids who don't want to go to school.
For example, the philosophers who were interested in logic were probably rather logical for mathematicians. But the ASL got us together, so we could talk to each other and publish in the same journal.
The introduction of the cipher 0 or the group concept was general nonsense too, and mathematics was more or less stagnating for thousands of years because nobody was around to take such childish steps.
Our minds are finite, and yet even in these circumstances of finitude we are surrounded by possibilities that are infinite, and the purpose of life is to grasp as much as we can out of that infinitude.
No period of history has ever been great or ever can be that does not act on some sort of high, idealistic motives, and idealism in our time has been shoved aside, and we are paying the penalty for it.
Vanity of science. Knowledge of physical science will not console me for ignorance of morality in time of affliction, but knowledge of morality will always console me for ignorance of physical science.
We feel neither extreme heat nor extreme cold; qualities that are in excess are so much at variance with our feelings that they are impalpable: we do not feel them, though we suffer from their effects.
Man's greatness is great in that he knows himself wretched. A tree does not know itself wretched. It is then being wretched to know oneself wretched; but it is being great to know that one is wretched.
Piety is different from superstition. To carry piety to the extent of superstition is to destroy it. The heretics reproach us with this superstitious submission. It is doing what they reproach us with.
The mind must not be forced; artificial and constrained manners fill it with foolish presumption, through unnatural elevation and vain and ridiculous inflation, instead of solid and vigorous nutriment.
As the sun eclipses the stars by its brilliancy, so the man of knowledge will eclipse the fame of others in assemblies of the people if he proposes algebraic problems, and still more if he solves them.
You should not put too much trust in any unproved conjecture, even if it has been propounded by a great authority, even if it has been propounded by yourself. You should try to prove it or disprove it.
I hold that the mark of a genuine idea is that its possibility can be proved, either a priori by conceiving its cause or reason, or a posteriori when experience teaches us that it is in fact in nature.
It consisteth in one knows not what, and springeth up one can hardly tell how. Its ways are unaccountable and inexplicable, being answerable to the numberless rovings of fancy and windings of language.
The mathematician requires tact and good taste at every step of his work, and he has to learn to trust to his own instinct to distinguish between what is really worthy of his efforts and what is not...
Every corporeal substance, so far forth as it is corporeal, has a natural fitness for resting in every place where it may be situated by itself beyond the sphere of influence of a body cognate with it.
The new atheism as an old atheism, except it's much more aggressive. The new atheism wants to destroy religion. That's a very different thing. The arguments are not new, it's the aggression that's new.
Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate.
Among all of the mathematical disciplines the theory of differential equations is the most important... It furnishes the explanation of all those elementary manifestations of nature which involve time.
A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories.
I believe that at the end of the century the use of words and general educated opinion will have altered so much that one will be able to speak of machines thinking without expecting to be contradicted.
But be that as it may, I think it is more respectful to you that I should speak to you upon and do my best to interest you in the subject which has occupied me, and in which I am myself most interested.
Humanity has known for a long time what fractals are. It is a very strange situation in which an idea which each time I look at all documents have deeper and deeper roots, never (how to say it), jelled.
What reason have atheists for saying that we cannot rise again? That what has never been, should be, or that what has been, should be again? Is it more difficult to come into being than to return to it.
Without [diversion] we would be in a state of weariness, and this weariness would spur us on to seek a more solid means of escaping from it. But diversion amuses us, and leads us unconsciously to death.
To praise it would amount to praising myself. For the entire content of the work... coincides almost exactly with my own meditations which have occupied my mind for the past thirty or thirty-five years.
To such idle talk it might further be added: that whenever a certain exclusive occupation is coupled with specific shortcomings, it is likewise almost certainly divorced from certain other shortcomings.
The half minute which we daily devote to the winding-up of our watches is an exertion of labour almost insensible; yet, by the aid of a few wheels, its effect is spread over the whole twenty-four hours.
Gravity is a mutual affection between cognate bodies towards union or conjunction (similar in kind to the magnetic virtue), so that the earth attracts a stone much rather than the stone seeks the earth.
Why are numbers beautiful? It’s like asking why is Beethoven’s Ninth Symphony beautiful. If you don’t see why, someone can’t tell you. I know numbers are beautiful. If they aren’t beautiful, nothing is.
Don't just read it; fight it! Ask your own question, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? ... Where does the proof use the hypothesis?
You can get a certain amount of pleasure as a mathematical spectator, reading and watching some of the most beautiful arguments that have been created in the history of humanity. But that's too passive.
It does not matter what men say in words, so long as their activities are controlled by settled instincts. The words may ultimately destroy the instincts; but until this has occurred, words do not count.
Systems, scientific or philosophic, come and go. Each method of limited understanding is at length exhausted. In its prime each system is a triumphant success: in its decay it is an obstructive nuisance.
Eureka! Eureka! Supposed to have been his cry, jumping naked from his bath and running in the streets, excited by a discovery about water displacement to solve a problem about the purity of a gold crown.
The theory of probability is the only mathematical tool available to help map the unknown and the uncontrollable. It is fortunate that this tool, while tricky, is extraordinarily powerful and convenient.
Our achievements of today are but the sum total of our thoughts of yesterday. You are today where the thoughts of yesterday have brought you and you will be tomorrow where the thoughts of today take you.
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.
The whole of the developments and operations of analysis are now capable of being executed by machinery ... As soon as an Analytical Engine exists, it will necessarily guide the future course of science.
I am inclined to attach some importance to the new system of manufacturing; and venture to throw it out with the hope of its receiving a full discussion among those who are most interestedin the subject.
The first rule of style is to have something to say. The second rule of style is to control yourself when, by chance, you have two things to say; say first one, then the other, not both at the same time.
There are also two kinds of truths: truth of reasoning and truths of fact. Truths of reasoning are necessary and their opposite is impossible; those of fact are contingent and their opposite is possible.
The time has mainly gone on getting Inform into a decent shape for public use. I suppose the plot of 'Curses' makes a sequel conceivable when compared with, say, the plot of 'Hamlet' but none is planned.
But it seems an irony of creation that man's mind knows how to handle things the better the farther removed they are from the center of his existence. Thus we are cleverest where knowledge matters least.
To those who ask what the infinitely small quantity in mathematics is, we answer that it is actually zero. Hence there are not so many mysteries hidden in this concept as they are usually believed to be.
This is really an opportunity to mobilize a new generation of scholars and pastors not just to equip the saints but also to engage the culture and reclaim it for Christ. That's really what is driving me.
Among the thousand-and-one faces whereby form chooses to reveal itself to us, the one that fascinates me more than any other, and continues to fascinate me, is the structure hidden in mathematical things.
Both parents worshiped individual achievement, but because of the Depression and the war, they never achieved what they wanted and deserved. So their ambition and high expectations were transferred to me.