Quotes of All Topics . Occasions . Authors
Every maker of video games knows something that the makers of curriculum don't seem to understand. You'll never see a video game being advertised as being easy. Kids who do not like school will tell you it's not because it's too hard. It's because it's--boring
Men despise religion. They hate it and are afraid it may be true. The cure for this is first to show that religion is not contrary to reason, but worthy of reverence and respect. Next make it attractive, make good men wish it were true and then show that it is.
[The] humanization of mathematical teaching, the bringing of the matter and the spirit of mathematics to bear not merely upon certain fragmentary faculties of the mind, but upon the whole mind, that this is the greatest desideratum is. I assume, beyond dispute.
It is hardly possible to maintain seriously that the evil done by science is not altogether outweighed by the good. For example, if ten million lives were lost in every war, the net effect of science would still have been to increase the average length of life.
He that loveth a book will never want a faithful friend, a wholesome counsellor, a cheerful companion, an effectual comforter. By study, by reading, by thinking, one may innocently divert and pleasantly entertain himself, as in all weathers, so in all fortunes.
The sense of tragedy is that the world is not a pleasant little nest made for our protection, but a vast and largely hostile environment, in which we can achieve great things only by defying the gods; and that this defiance inevitably brings its own punishment.
In order to turn natural history into a true science, one would have to devote oneself to investigations capable of telling us not the particular shape of such and such an animal, but the general procedures of nature in the animal's production and preservation.
The misconception which has haunted philosophic literature throughout the centuries is the notion of 'independent existence.' There is no such mode of existence; every entity is to be understood in terms of the way it is interwoven with the rest of the universe.
I conceived and developed a new geometry of nature and implemented its use in a number of diverse fields. It describes many of the irregular and fragmented patterns around us, and leads to full-fledged theories, by identifying a family of shapes I call fractals.
Il n'y a que deux sortes d'hommes: les uns justes, qui se croient pe cheurs; les autres pe cheurs, qui se croient justes. There are only two types of people: the virtuous who believe themselves to be sinners and the sinners who believe themselves to be virtuous.
The Church has been reproached with endeavouring to appropriate to itself all those professorships in our Universities which are connected with science: it is however certain that the larger portion of these ill-remunerated offices have been filled by clergymen.
Mathematics are the result of mysterious powers which no one understands, and which the unconscious recognition of beauty must play an important part. Out of an infinity of designs a mathematician chooses one pattern for beauty's sake and pulls it down to earth.
Repetition acts as an enforcement mechanism: It makes cooperation achievable when it is not achievable in the one-shot game, even when one replaces strategic equilibrium as the criterion for achievability by the more stringent requirement of perfect equilibrium.
The standard "foundation" for mathematics starts with sets and their elements. It is possible to start differently, by axiomatising not elements of sets but functions between sets. This can be done by using the language of categories and universal constructions.
The triumph of the industrial arts will advance the cause of civilization more rapidly than its warmest advocates could have hoped, and contribute to the permanent prosperity and strength of the country far more than the most splendid victories of successful war.
This view [of the infinite], which I consider to be the sole correct one, is held by only a few. While possibly I am the very first in history to take this position so explicitly, with all of its logical consequences, I know for sure that I shall not be the last!
The idea that information can be stored in a changing world without an overwhelming depreciation of its value is false. It is scarcely less false than the more plausible claim that after a war we may take our existing weapons, fill their barrels with information.
We conclude that, simultaneously with the organization of the colleges, there should be at Santa Cruz an organization by disciplines, whose units would have a voice in appointments and promotions, in course of programs, and in the allocation of funds for research.
Imagine a person with a gift of ridicule [He might say] First that a negative quantity has no logarithm; secondly that a negative quantity has no square root; thirdly that the first non-existent is to the second as the circumference of a circle is to the diameter.
The greatness of man is so evident that it is even proved by his wretchedness. For what in animals is nature, we call in man wretchedness--by which we recognize that, his nature being now like that of animals, he has fallen from a better nature which once was his.
Whilst in speaking of human things, we say that it is necessary to know them before we love can them. The saints on the contrary say in speaking of divine things that it is necessary to love them in order to know them, and that we only enter truth through charity.
Fuller believed human societies would soon rely mainly on renewable sources of energy, such as solar- and wind-derived electricity,. envisioned an age of "universal education and sustenance of all humanity". "The heart has reasons that reason does not understand."
The tastes and pursuits of manhood will bear on them the traces of the earlier impressions of our education. It is therefore not unreasonable to suppose that some portion of the neglect of science in England, may be attributed to the system of education we pursue.
Every one who understands the subject will agree that even the basis on which the scientific explanation of nature rests is intelligible only to those who have learned at least the elements of the differential and integral calculus, as well as analytical geometry.
All existing things upon this earth, which have knowledge of their own existence, possess, some in one degree and some in another, the power of thought, accompanied by perception, which is the awakening of thought by the effects of external objects upon the senses.
A mathematical problem should be difficult in order to entice us, yet not completely inaccessible, lest it mock at our efforts. It should be to us a guide post on the mazy paths to hidden truths, and ultimately a reminder of our pleasure in the successful solution.
... if one were to refuse to have direct, geometric, intuitive insights, if one were reduced to pure logic, which does not permit a choice among every thing that is exact, one would hardly think of many questions, and certain notions ... would escape us completely.
Even as the finite encloses an infinite series And in the unlimited limits appear, So the soul of immensity dwells in minutia And in the narrowest limits no limit in here. What joy to discern the minute in infinity! The vast to perceive in the small, what divinity!
But just as much as it is easy to find the differential [derivative] of a given quantity, so it is difficult to find the integral of a given differential. Moreover, sometimes we cannot say with certainty whether the integral of a given quantity can be found or not.
It's not what you know about the computer that's important, but your ability to do things with it. By studying French in an academic setting, you get to know a lot about it, but typically, you can't express yourself well or have an interesting conversation with it.
Kind words produce their own image in men's souls; and a beautiful image it is. They soothe and quiet and comfort the hearer. They shame him out of his sour, morose, unkind feelings. We have not yet begun to use kind words in such abundance as they ought to be used.
When I have occasionally set myself to consider the different distractions of men, the pains and perils to which they expose themselves I have discovered that all the unhappiness of men arises from one single fact, that they cannot stay quietly in their own chamber.
What matters it that man should have a little more knowledge of the universe? If he has it, he gets little higher. Is he not always infinitely removed from the end, and is not the duration of our life equally removed from eternity, even if it lasts ten years longer?
Pure mathematics offers no mercenary inducements to its followers, who is attracted to it by the importance and beauty of the truths in contains; and the complete absence of any material advantage to be gained by means of it, adds perhaps another charm to its study.
Some atheists are quite explicit that their atheism comes first. One of the most famous is Richard Lewontin, a professor of genetics, who said it wasn't science that compelled him to accept a materialistic explanation of the universe. It was an a priori materialism.
Does anyone believe that the difference between the Lebesgue and Riemann integrals can have physical significance, and that whether say, an airplane would or would not fly could depend on this difference? If such were claimed, I should not care to fly in that plane.
A typical mathematician does not actively try to be useful. Individual mathematicians are motivated primarily by a subtle mixture of ambition and intellectual curiosity, and not by a wish to benefit society, nevertheless, mathematics as a whole does benefit society.
Having been the discoverer of many splendid things, he is said to have asked his friends and relations that, after his death, they should place on his tomb a cylinder enclosing a sphere, writing on it the proportion of the containing solid to that which is contained.
I've been a professor of mathematics at Harvard and at Yale. At Yale for a long time. But I'm not a mathematician only. I'm a professor of physics, of economics, a long list. Each element of this list is normal. The combination of these elements is very rare at best.
The true method of discovery is like the flight of an aeroplane. It starts from the ground of particular observation; it makes a flight in the thin air of imaginative generalization; and it again lands for renewed observation rendered acute by rational interpretation.
But we must not underestimate the potency of the mathematical process of abstraction. A surprising variety of things happen to have both magnitude and direction and to combine according to the parallelogram law; and many of them are not at all reminiscent of journeys.
Philosophers and psychiatrists should explain why it is that we mathematicians are in the habit of systematically erasing our footsteps. Scientists have always looked askance at this strange habit of mathematicians, which has changed little from Pythagoras to our day.
It may be appropriate to quote a statement of Poincare, who said (partly in jest no doubt) that there must be something mysterious about the normal law since mathematicians think it is a law of nature whereas physicists are convinced that it is a mathematical theorem.
Those who know that the consensus of many centuries has sanctioned the conception that the earth remains at rest in the middle of the heavens as its center, would, I reflected, regard it as an insane pronouncement if I made the opposite assertion that the earth moves.
I was still very hopeful that much work lay ahead of me. Perhaps because much of what I had worked on or thought about had not yet been put into writing, I felt I still had things in reserve. Given this optimistic nature, I feel this way even now when I am past sixty.
Einstein does not remain attached to the classical principles, and when presented with a problem in physics he quickly envisages all of its possibilities. This leads immediately in his mind to the prediction of new phenomena which may one day be verified by experiment.
It is a misfortune for a science to be born too late when the means of observation have become too perfect. That is what is happening at this moment with respect to physical chemistry; the founders are hampered in their general grasp by third and fourth decimal places.
Science is a very human form of knowledge. We are always at the brink of the known; we always feel forward for what is to be hoped. Every judgment in science stands on the edge of error and is personal. Science is a tribute to what we can know although we are fallible.
The theory of ramification is one of pure colligation, for it takes no account of magnitude or position; geometrical lines are used, but these have no more real bearing on the matter than those employed in genealogical tables have in explaining the laws of procreation.
The classification of facts, the recognition of their sequence and relative significance is the function of science, and the habit of forming a judgment upon these facts unbiassed by personal feeling is characteristic of what may be termed the scientific frame of mind.