Quotes of All Topics . Occasions . Authors
When a natural discourse paints a passion or an effect, one feels within oneself the truth of what one reads, which was there before, although one did not know it. Hence one is inclined to love him who makes us feel it, for he has not shown us his own riches, but ours. ...such community of intellect that we have with him necessarily inclines the heart to love.
Although the semicircle of the Moon is placed above the circle of the Sun and would appear to be superior, nevertheless we know that the Sun is ruler and King. We see that the Moon in her shape and her proximity rivals the Sun with her grandeur, which is apparent to ordinary men, yet the face, or a semi-sphere of the Moon, always reflects the light of the Sun.
One might say the computer is being used to program the child. In my vision, the child programs the computer, and in doing so, both acquires a sense of mastery over a piece of the most modern and powerful technology and establishes an intense contact with some of the deepest ideas from science, from mathematics, and from the art of intellectual model building.
As we speak of poetical beauty, so ought we to speak of mathematical beauty and medical beauty. But we do not do so; and that reason is that we know well what is the object of mathematics, and that it consists in proofs, and what is the object of medicine, and that it consists in healing. But we do not know in what grace consists, which is the object of poetry.
[On Sophie Germain] When a person of the sex which, according to our customs and prejudices, must encounter infinitely more difficulties than men... succeeds nevertheless in surmounting these obstacles and penetrating the most obscure parts of [number theory], then without doubt she must have the noblest courage, quite extraordinary talents and superior genius.
There are two famous labyrinths where our reason very often goes astray. One concerns the great question of the free and the necessary, above all in the production and the origin of Evil. The other consists in the discussion of continuity, and of the indivisibles which appear to be the elements thereof, and where the consideration of the infinite must enter in.
So long as a man remains a gregarious and sociable being, he cannot cut himself off from the gratification of the instinct of imparting what he is learning, of propagating through others the ideas and impressions seething in his own brain, without stunting and atrophying his moral nature and drying up the surest sources of his future intellectual replenishment.
The Ancients, having taken into consideration the rigorous construction of the human body, elaborated all their works, as especially their holy temples, according to these proportions; for they found here the two principal figures without which no project is possible: the perfection of the circle, the principle of all regular bodies, and the equilateral square.
The art of Navigation demonstrates how, by the shortest good way, by the aptest direction, and in the shortest time, a sufficient ship, between any two places (in passage navigable) assigned, may be conducted; and in all storms and natural disturbances chancing, how to use the best possible means, whereby to recover the place first assigned. Mathematical Preface
If they [Plato and Aristotle] wrote about politics it was as if to lay down rules for a madhouse. And if they pretended to treat it as something really important it was because they knew that the madmen they were talking to believed themselves to be kings and emperors. They humored these beliefs in order to calm down their madness with as little harm as possible.
Knowledge has two extremes. The first is the pure natural ignorance in which all men find themselves at birth. The other extreme is that reached by great minds, who, having run through all that men can know, find they know nothing, and come back again to that same natural ignorance from which they set out; this is a learned ignorance which is conscious of itself.
My work on prime gaps lead to lots of media coverage, some good, some bad, some ugly, and some merely ridiculous. For example, a reporter of our university newspaper, who admitted that he is still learning English, wrote that "Prof. Goldston solved one of the most controversial problems in the prime number theory last month with support from his Turkish partner."
If a man is in any sense a real mathematician, then it is a hundred to one that his mathematics will be far better than anything else he can do, and that he would be silly if he surrendered any decent opportunity of exercising his one talent in order to do undistinguished work in other fields. Such a sacrifice could be justified only by economic necessity of age.
The attempt to apply rational arithmetic to a problem in geometry resulted in the first crisis in the history of mathematics. The two relatively simple problems -- the determination of the diagonal of a square and that of the circumference of a circle -- revealed the existence of new mathematical beings for which no place could be found within the rational domain.
If there is one thing in mathematics that fascinates me more than anything else (and doubtless always has), it is neither "number" nor "size", but always form. And 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.
I presume that to the uninitiated the formulae will appear cold and cheerless; but let it be remembered that, like other mathematical formulae, they find their origin in the divine source of all geometry. Whether I shall have the satisfaction of taking part in their exposition, or whether that will remain for some more profound expositor, will be seen in the future.
Let us, then, take our compass; we are something, and we are not everything. The nature of our existence hides from us the knowledge of first beginnings which are born of the nothing; and the littleness of our being conceals from us the sight of the infinite. Our intellect holds the same position in the world of thought as our body occupies in the expanse of nature.
Mathematics is not arithmetic. Though mathematics may have arisen from the practices of counting and measuring it really deals with logical reasoning in which theorems-general and specific statements-can be deduced from the starting assumptions. It is, perhaps, the purest and most rigorous of intellectual activities, and is often thought of as queen of the sciences.
Tests showed cancer of the larynx and the doctor advised an operation immediately. I was informed that my larynx had to be removed completely. I heard about Dr Breuss and went to see him....he prescribed the juice treatment....By the time I had completed this juice treatment I felt fit and once again had a good appetite. Despite my 72 years I felt my old self again.
To throw in a fair game at Hazards only three-spots, when something great is at stake, or some business is the hazard, is a natural occurrence and deserves to be so deemed; and even when they come up the same way for a second time if the throw be repeated. If the third and fourth plays are the same, surely there is occasion for suspicion on the part of a prudent man.
I've got a telescope in my garden and one of the things I love to do is go out and let the sky, the night sky, the galaxies, the Orion nebula, have an impact on my mind. I find that awe inspiring. And just to contemplate on what the astronomers have revealed to us about the immense size and so on of the universe. I find that very healthy. And it's a good thing to do.
Some people think that mathematics is a serious business that must always be cold and dry; but we think mathematics is fun, and we aren't ashamed to admit the fact. Why should a strict boundary line be drawn between work and play? Concrete mathematics is full of appealing patterns; the manipulations are not always easy, but the answers can be astonishingly attractive.
The vitality of thought is in adventure. Idea's won't keep. Something must be done about them. When the idea is new, its custodians have fervour, live for it, and, if need be, die for it. Their inheritors receive the idea, perhaps now strong and successful, but without inheriting the fervour; so the idea settles down to a comfortable middle age, turns senile, and dies.
That queen, of error, whom we call fancy and opinion, is the more deceitful because she does not always deceive. She would be the infallible rule of truth if she were the infallible rule of falsehood; but being only most frequently in error, she gives no evidence of her real quality, for she marks with the same character both that which is true and that which is false.
What I assert and believe to have demonstrated in this and earlier works is that following the finite there is a transfinite (which one could also call the supra-finite), that is an unbounded ascending lader of definite modes, which by their nature are not finite but infinite, but which just like the finite can be determined by well-defined and distinguishable numbers.
If you have to prove a theorem, do not rush. First of all, understand fully what the theorem says, try to see clearly what it means. Then check the theorem; it could be false. Examine the consequences, verify as many particular instances as are needed to convince yourself of the truth. When you have satisfied yourself that the theorem is true, you can start proving it.
When every fact, every present or past phenomenon of that universe, every phase of present or past life therein, has been examined, classified, and co-ordinated with the rest, then the mission of science will be completed. What is this but saying that the task of science can never end till man ceases to be, till history is no longer made, and development itself ceases?
When, therefore, I had long considered this uncertainty of traditional mathematics, it began to weary me that no more definite explanation of the movement of the world-machine established in our behalf by the best and most systematic builder of all, existed among the philosophers who had studied so exactly in other respects the minutest details in regard to the sphere.
Il y a deux sortes d'esprits, l'un ge ome trique, et l'autre que l'on peut appeler de finesse. Le premier a des vues lentes, dures et inflexibles; mais le dernier a une souplesse de pense e. There are two kinds of mind, one mathematical, the other what one might call the intuitive. The first takes a slow, firm, inflexible view, but the latter has flexibility of thought.
In my opinion instruction is very purposeless for such individuals who do no want merely to collect a mass of knowledge, but are mainly interested in exercising (training) their own powers. One doesn't need to grasp such a one by the hand and lead him to the goal, but only from time to time give him suggestions, in order that he may reach it himself in the shortest way.
Indeed, the most important part of engineering work-and also of other scientific work-is the determination of the method of attacking the problem, whatever it may be, whether an experimental investigation, or a theoretical calculation. ... It is by the choice of a suitable method of attack, that intricate problems are reduced to simple phenomena, and then easily solved.
All men seek happiness. This is without exception. Whatever different means they employ, they all tend to this end. The cause of some going to war, and of others avoiding it, is the same desire in both, attended with different views. The will never takes the least step but to this object. This is the motive of every action of every man, even of those who hang themselves.
The man who knows God but does not know his own misery, becomes proud. The man who knows his own misery but does not know God, ends in despair...the knowledge of Jesus Christ constitutes the middle course because in him we find both God and our own misery. Jesus Christ is therefore a God whom we approach without pride, and before whom we humble ourselves without despair.
Catastrophe Theory is-quite likely-the first coherent attempt (since Aristotelian logic) to give a theory on analogy. When narrow-minded scientists object to Catastrophe Theory that it gives no more than analogies, or metaphors, they do not realise that they are stating the proper aim of Catastrophe Theory, which is to classify all possible types of analogous situations.
In the history of the world the prize has not gone to those species which specialized in methods of violence, or even in defensive armor. In fact, nature began with producing animals encased in hard shells for defense against the ill of life. But smaller animals, without external armor, warm-blooded, sensitive, alert, have cleared those monsters off the face of the earth.
The practical man demands an appearance of reality at least. Always dealing in the concrete, he regards mathematical terms not as symbols or thought but as images of reality. A system acceptable to the mathematician because of its inner consistency may appear to the practical man to be full of contradictions because of the incomplete manner in which it represents reality.
Man is full of desires: he loves only those who can satisfy them all. "This man is a good mathematician," someone will say. But I have no concern for mathematics; he would take me for a proposition. "That one is a good soldier." He would take me for a besieged town. I need, that is to say, a decent man who can accommodate himself to all my desires in a general sort of way.
The presentation of mathematics in schools should be psychological and not systematic. The teacher, so to speak, should be a diplomat. He must take account of the psychic processes in the boy in order to grip his interest, and he will succeed only if he presents things in a form intuitively comprehensible. A more abstract presentation is only possible in the upper classes.
Either... the moving intelligences of the planets are weakest in those that are farthest from the sun, or... there is one moving intelligence in the sun, the common center, forcing them all round, but those most violently which are nearest, and that it languishes in some sort and grows weaker at the most distant, because of the remoteness and the attenuation of the virtue.
How then shall mathematical concepts be judged? They shall not be judged. Mathematics is the supreme arbiter. From its decisions there is no appeal. We cannot change the rules of the game, we cannot ascertain whether the game is fair. We can only study the player at his game; not, however, with the detached attitude of a bystander, for we are watching our own minds at play.
In every science, after having analysed the ideas, expressing the more complicated by means of the more simple, one finds a certain number that cannot be reduced among them, and that one can define no further. These are the primitive ideas of the science; it is necessary to acquire them through experience, or through induction; it is impossible to explain them by deduction.
When we find ourselves unable to reason (as one often does when presented with, say, a problem in algebra) it is because our imagination is not touched. One can begin to reason only when a clear picture has been formed in the imagination. Bad teaching is teaching which presents an endless procession of meaningless signs, words and rules, and fails to arouse the imagination.
In England if something goes wrong--say, if one finds a skunk in the garden--he writes to the family solicitor, who proceeds to take the proper measures; whereas in America, you telephone the fire department. Each satisfies a characteristic need; in the English, love of order and legalistic procedure; and here in America, what you like is something vivid, and red, and swift.
I presume that few who have paid any attention to the history of the Mathematical Analysis, will doubt that it has been developed in a certain order, or that that order has been, to a great extent, necessary -- being determined, either by steps of logical deduction, or by the successive introduction of new ideas and conceptions, when the time for their evolution had arrived.
No science is immune to the infection of politics and the corruption of power. ... The time has come to consider how we might bring about a separation, as complete as possible, between Science and Government in all countries. I call this the disestablishment of science, in the same sense in which the churches have been disestablished and have become independent of the state.
Until now the theory of infinite series in general has been very badly grounded. One applies all the operations to infinite series as if they were finite; but is that permissible? I think not. Where is it demonstrated that one obtains the differential of an infinite series by taking the differential of each term? Nothing is easier than to give instances where this is not so.
[The works of Archimedes] are without exception, monuments of mathematical exposition; the gradual revelation of the plan of attack, the masterly ordering of the propositions, the stern elimination of everything not immediately relevant to the purpose, the finish of the whole, are so impressive in their perfection as to create a feeling akin to awe in the mind of the reader.
The manner of Demoivre's death has a certain interest for psychologists. Shortly before it, he declared that it was necessary for him to sleep some ten minutes or a quarter of an hour longer each day than the preceding one: the day after he had thus reached a total of something over twenty-three hours he slept up to the limit of twenty-four hours, and then died in his sleep.
I had made considerable advance ... in calculations on my favourite numerical lunar theory, when I discovered that, under the heavy pressure of unusual matters (two transits of Venus and some eclipses) I had committed a grievous error in the first stage of giving numerical value to my theory. My spirit in the work was broken, and I have never heartily proceeded with it since.
He who will please the crowd and for the sake of the most ephemeral renown will either proclaim those things which nature does not display or even will publish genuine miracles of nature without regard to deeper causes is a spiritually corrupt person... With the best of intentions I publicly speak to the crowd (which is eager for things new) on the subject of what is to come.