Quotes of All Topics . Occasions . Authors
A modern librarian, who has faith in the law that 'BOOKS ARE FOR USE,' is happy only when his readers make his shelves constantly empty. It is not the books that go out that worry him. It is the stay-at-home volumes that perplex and depress him.
When a branch of mathematics ceases to interest any but the specialists, it is very near its death, or at any rate dangerously close to a paralysis, from which it can be rescued only by being plunged back into the vivifying source of the science.
We are so presumptuous that we should like to be known all over the world, even by people who will only come when we are no more. Such is our vanity that the good opinion of half a dozen of the people around us gives us pleasure and satisfaction.
The whole is always more, is more capable of a much greater variety of wave states, than the combination of its parts. ... In this very radical sense, quantum physics supports the doctrine that the whole is more than the combination of its parts.
That justice should be administered between men, it is necessary that testimonies of fact be alleged; and that witnesses should apprehend themselves greatly obliged to discover the truth, according to their conscience, in dark and doubtful cases.
It has always puzzled me that so many people have taken it for granted that God favors those who believe in him. Isn't it possible that the actual God is a scientific God who has little patience with beliefs founded on faith rather than evidence?
Round about the accredited and orderly facts of every science there ever floats a sort of dustcloud of exceptional observations, of occurrences minute and irregular and seldom met with, which it always proves more easy to ignore than to attend to.
The teacher manages to get along still with the cumbersome algebraic analysis, in spite of its difficulties and imperfections, and avoids the smooth infinitesimal calculus, although the eighteenth century shyness toward it had long lost all point.
I also readily admit that there are animals, taken in the ordinary sense, that are incomparably larger than those we know of, and I have sometimes said in jest that there might be a system like ours which is the pocketwatch of some enormous giant.
Music is part of Number Theory. Nowadays when a number-theorist applies for a grant, he says that it is good for security, but in those days, way before America, he would say that it's good for music. I will not comment whether we have progressed.
The modern physicist is a quantum theorist on Monday, Wednesday, and Friday, and a student of gravitational relativity on Tuesday, Thursday, and Saturday. On Sunday, he is praying... that someone will find the reconciliation between the two views.
One feature which will probably most impress the mathematician accustomed to the rapidity and directness secured by the generality of modern methods is the deliberation with which Archimedes approaches the solution of any one of his main problems.
All men are almost led to believe not of proof, but by attraction. This way is base, ignoble, and irrelevant; every one therefore disavows it. Each one professes to believe and even to love nothing but what he knows to be worthy of belief and love.
Dream or nightmare, we have to live our experience as it is, and we have to live it awake. We live in a world which is penetrated through and through by science and which is both whole and real. We cannot turn it into a game simple by taking sides.
In passing, I firmly believe that research should be offset by a certain amount of teaching, if only as a change from the agony of research. The trouble, however, I freely admit, is that in practice you get either no teaching, or else far too much.
One of the greatest artifices the devil uses to engage men in vice and debauchery, is to fasten names of contempt on certain virtues, and thus fill weak souls with a foolish fear of passing for scrupulous, should they desire to put them in practice.
No person will deny that the highest degree of attainable accuracy is an object to be desired, and it is generally found that the last advances towards precision require a greater devotion of time, labour, and expense, than those which precede them.
The accumulation of skill and science which has been directed to diminish the difficulty of producing manufactured goods, has not been beneficial to that country alone in which it is concentrated distant kingdoms have participated in its advantages.
My own view is that no major advance was ever found as a result of a committee's recommendation... We should be honest in telling these agencies we often don't know where some ideas are going to lead, but we hope they are going to clarify a problem.
All that is not thought is pure nothingness; since we can think only thoughts, and all the words we use to speak of things can express only thoughts, to say there is something other than thought is therefore an affirmation which can have no meaning.
Isolated, so-called "pretty theorems" have even less value in the eyes of a modern mathematician than the discovery of a new "pretty flower" has to the scientific botanist, though the layman finds in these the chief charm of the respective sciences.
Whenever a text-book is written of real educational worth, you may be quite certain that some reviewer will say that it will be difficult to teach from it. Of course it will be difficult to teach from it. It it were easy, the book ought to be burned.
Self-similarity is a dull subject because you are used to very familiar shapes. But that is not the case. Now many shapes which are self-similar again, the same seen from close by and far away, and which are far from being straight or plane or solid.
The accumulation of skill and science which has been directed to diminish the difficulty of producing manufactured goods, has not been beneficial to that country alone in which it is concentrated; distant kingdoms have participated in its advantages.
The mathematician's patterns, like the painter's or the poet's must be beautiful; the ideas, like the colours or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.
Probability is expectation founded upon partial knowledge. A perfect acquaintance with all the circumstances affecting the occurrence of an event would change expectation into certainty, and leave nether room nor demand for a theory of probabilities.
Eventually I found it had been working all along-but didn't show anything on screen until it had the first full page of text. I inserted 30 new lines, and suddenly my toy said 'hEllO woRlD'. An hour later I understood alphabet shifting rather better!
We imagine a school in which students and teachers excitedly and joyfully stretch themselves to their limits in pursuit of projects built on their own visions … not one that that merely succeeds in making apathetic students satisfy minimal standards.
True eloquence makes light of eloquence, true morality makes light of morality; that is to say, the morality of the judgment, which has no rules, makes light of the morality of the intellect.... To make light of philosophy is to be a true philosopher.
One hears a lot of talk about the hostility between scientists and engineers. I don't believe in any such thing. In fact I am quite certain it is untrue... There cannot possibly be anything in it because neither side has anything to do with the other.
Hilbert once had a student in mathematics who stopped coming to his lectures, and he was finally told the young man had gone off to become a poet. Hilbert is reported to have remarked: 'I never thought he had enough imagination to be a mathematician.'
If we desire to live securely, comfortably, and quietly, that by all honest means we should endeavor to purchase the good will of all men, and provoke no man's enmity needlessly; since any man's love may be useful, and every man's hatred is dangerous.
Mathematics, in the common lay view, is a static discipline based on formulas...But outside the public view, mathematics continues to grow at a rapid rate...the guid to this growth is not calculation and formulas, but an open ended search for pattern.
Therefore I would not have it unknown to Your Holiness, the the only thing which induced me to look for another way of reckoning the movements of the heavenly bodies was that I knew that mathematicians by no means agree in their investigation thereof.
Often kids in a computer lab learn about word-processing, but if they want to write an essay, they write it by hand. This is exactly the opposite of what you want them to learn. They're approaching the computer as just another abstract school subject.
By explanation the scientist understands nothing except the reduction to the least and simplest basic laws possible, beyond which he cannot go, but must plainly demand them; from them however he deduces the phenomena absolutely completely as necessary.
Mathematics is the study of analogies between analogies. All science is. Scientists want to show that things that don't look alike are really the same. That is one of their innermost Freudian motivations. In fact, that is what we mean by understanding.
It may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the later. Prediction becomes impossible, and we have the fortuitous phenomena.
The year 1915 was one of meager results, the advantages remaining on the side of the Central Powers, with this understanding, however: The Allies were growing stronger because Great Britain was making rapid progress in marshaling her resources for war.
Eloquence is a way of saying things in such a way, first, that those to whom we speak may listen to them without pain and with pleasure, and second, that they feel themselves interested, so that self-love leads them more willingly to reflection upon it.
From a mathematical standpoint it is possible to have infinite space. In a mathematical sense space is manifoldness, or combinations of numbers. Physical space is known as the 3-dimension system. There is the 4-dimension system, the 10-dimension system.
What humans do with the language of mathematics is to describe patterns... To grow mathematically children must be exposed to a rich variety of patterns appropriate to their own lives through which they can see variety, regularity, and interconnections.
The joy of suddenly learning a former secret and the joy of suddenly discovering a hitherto unknown truth are the same to me - both have the flash of enlightenment, the almost incredibly enhanced vision, and the ecstasy and euphoria of released tension.
The scientific picture of the world championed since the Enlightenment is not just wrong but massively wrong. Indeed entire fields of inquiry, especially in the human sciences, will need to be rethought from the ground up in terms of intelligent design.
The point about zero is that we do not need to use it in the operations of daily life. No one goes out to buy zero fish. It is in a way the most civilised of all the cardinals, and its use is only forced on us by the needs of cultivated modes of thought.
Great fleas have little fleas upon their backs to bite 'em, And little fleas have lesser fleas, and so ad infinitum, And the great fleas themselves, in turn, have greater fleas to go on, While these again have greater still, and greater still, and so on.
When I first began studying prices, it wasn't a topic that mathematicians were working on. Purely by accident, I saw a set of data on price changes presented in a lecture and realized they behaved similarly to the geometric models I was already studying.
Nature imitates herself. A grain thrown into good ground brings forth fruit; a principle thrown into a good mind brings forth fruit. Everything is created and conducted by the same Master-the root, the branch, the fruits-the principles, the consequences.
There are hardly any truths upon which we always remain agreed, and still fewer objects of pleasure which we do not change every hour, I do not know whether there is a means of giving fixed rules for adapting discourse to the inconstancy of our caprices.
I am coming more and more to the conviction that the necessity of our geometry cannot be demonstrated, at least neither by, nor for, the human intellect. . . Geometry should be ranked, not with arithmetic, which is purely aprioristic, but with mechanics.