Quotes of All Topics . Occasions . Authors
Although to penetrate into the intimate mysteries of nature and thence to learn the true causes of phenomena is not allowed to us, nevertheless it can happen that a certain fictive hypothesis may suffice for explaining many phenomena.
Accordingly, since nothing prevents the earth from moving, I suggest that we should now consider also whether several motions suit it, so that it can be regarded as one of the planets. For, it is not the center of all the revolutions.
I can readily conceive of a man without hands or feet; and I could conceive of him without a head, if experience had not taught me that by this he thinks, Thought then, is the essence of man, and without this we cannot conceive of him.
I take it as a matter not to be disputed, that if all knew what each said of the other, there would not be four friends in the world. This seems proved by the quarrels and disputes caused by the disclosures which are occasionally made.
There are certainly people who regard √2 as something perfectly obvious but jib at √-1. This is because they think they can visualise the former as something in physical space but not the latter. Actually √-1 is a much simpler concept.
There is a famous formula, perhaps the most compact and famous of all formulas - developed by Euler from a discovery of de Moivre: e^(i pi) + 1 = 0... It appeals equally to the mystic, the scientist, the philosopher, the mathematician.
I entertain no doubts as to the truths of the tranfinites, which I recognized with God's help and which, in their diversity, I have studied for more than twenty years; every year, and almost every day brings me further in this science.
A scientist can hardly meet with anything more undesirable than to have the foundations give way just as the work is finished. I was put in this position by a letter from Mr. Bertrand Russell when the work was nearly through the press.
Whence it is somewhat strange that any men from so mean and silly a practice should expect commendation, or that any should afford regard thereto; the which it is so far from meriting, that indeed contempt and abhorrence are due to it.
The most vitally characteristic fact about mathematics is, in my opinion, its quite peculiar relationship to the natural sciences, or more generally, to any science which interprets experience on a higher than purely descriptive level.
The reader will find no figures in this work. The methods which I set forth do not require either constructions or geometrical or mechanical reasonings: but only algebraic operations, subject to a regular and uniform rule of procedure.
I certainly do care about measuring educational results. But what is an 'educational result?' The twinkling eyes of my students, together with their heartfelt and beautifully expressed mathematical arguments are all the results I need.
A mathematician would hardly call a correspondence between the set of 64 triples of four units and a set of twenty other units, "universal", while such correspondence is, probably, the most fundamental general feature of life on Earth.
If the system breaks down the consequences will still be very painful. But the bigger the system grows the more disastrous the results of its breakdown will be, so if it is to break down it had best break down sooner rather than later.
To go beyond the bounds of moderation is to outrage humanity. The greatness of the human soul is shown by knowing how to keep within proper bounds. There are two equally dangerous extremes- to shut reason out, and not to let nothing in.
The pursuit of pretty formulas and neat theorems can no doubt quickly degenerate into a silly vice, but so can the quest for austere generalities which are so very general indeed that they are incapable of application to any particular.
If the proof starts from axioms, distinguishes several cases, and takes thirteen lines in the text book ... it may give the youngsters the impression that mathematics consists in proving the most obvious things in the least obvious way.
Making mathematics accessible to the educated layman, while keeping high scientific standards, has always been considered a treacherous navigation between the Scylla of professional contempt and the Charybdis of public misunderstanding.
Geometric calculus consists in a system of operations analogous to those of algebraic calculus, but in which the entities on which the calculations are carried out, instead of being numbers, are geometric entities which we shall define.
There never is absolute birth nor complete death, in the strict sense, consisting in the separation of the soul from the body. What we call births are developments and growths, while what we call deaths are envelopments and diminutions.
I grew up in a family with three siblings. My parents were always very supportive and encouraging. It was important for them that we have meaningful and satisfying professions, but they didn't care as much about success and achievement.
If there be some who, though ignorant of all mathematics . . . dare to reprove this work, because of some passage of Scripture, which they have miserably warped to their purpose, I regard them not, and even despise their rash judgement.
Because I have been a magician for many years, people have often asked me whether I ever have sawn a woman in half. I reply, Oh, yes I've sawn over seventy women in half in my lifetime, and I'm learning the second half of the trick now.
Even the simplest calculation in the purest mathematics can have terrible consequences. Without the invention of the infinitesimal calculus most of our technology would have been impossible. Should we say therefore that calculus is bad?
The problem of good as it faces the atheist is this: Nature, which is the nuts-and-bolts reality for the atheist, has no values and thus can offer no grounding for good and evil. Values on the atheist view are subjective and contingent.
Namely, we have no right to believe a thing true because everybody says so unless there are good grounds for believing that some one person at least has the means of knowing what is true, and is speaking the truth so far as he knows it.
If one takes the kinds of risks which I took, which are colossal, but taking risks, I was rewarded by being able to contribute in a very substantial fashion to a variety of fields. I was able to reawaken and solve some very old problems.
Not only do we know God by Jesus Christ alone, but we know ourselves only by Jesus Christ. We know life and death only through Jesus Christ. Apart from Jesus Christ, we do not know what is our life, nor our death, nor God, nor ourselves.
Quand on voit le style naturel, on est tout e tonne et ravi, car on s'attendait de voir un auteur, et on trouve un homme. When we see a natural style we are quite amazed and delighted, because we expected to see an author and find a man.
Things have different qualities, and the soul different inclinations; for nothing is simple which is presented to the soul, and the soul never presents itself simply to any object. Hence it comes that we weep and laugh at the same thing.
We shall be less apt to admire what this World calls Great, shall nobly despise those Trifles the generality of Men set their Affections on, when we know that there are a multitude of such Earths inhabited and adorned as Well as our own.
No mathematician of equal stature has risen from our generation... Hilbert was singularly free from national and racial prejudices; in all public questions, be they political, social or spiritual, he stood forever on the side of freedom.
Peace appeals to the hearts; studies to the brain. Both are needed, indeed indispensable. But equally indispensable is a valid link between brain and heart. And that, in a nutshell, is what peace studies and peace practice are all about.
The completion of a rigorous course in mathematics - it is not even necessary that the student does well in such a course - appears to be an excellent means of sharpening the mind and developing mental skills that are of general benefit.
The word 'chance' then expresses only our ignorance of the causes of the phenomena that we observe to occur and to succeed one another in no apparent order. Probability is relative in part to this ignorance, and in part to our knowledge.
Islamic terrorists are the greatest security threats to the United States, both domestically and abroad. We need to be vigilant every day. Now, every citizen has to be vigilant because they are vital links in the national security chain.
The product of mathematics is clarity and understanding. Not theorems, by themselves. ... In short, mathematics only exists in a living community of mathematicians that spreads understanding and breathes life into ideas both old and new.
For three days now this angel, almost too heavenly for earth has been my fiancée ... Life stands before me like an eternal spring with new and brilliant colours. Upon his engagement to Johanne Osthof of Brunswick; they married 9 Oct 1805.
In England, the profession of the law is that which seems to hold out the strongest attraction to talent, from the circumstance, that in it ability, coupled with exertion, even though unaided by patronage, cannot fail of obtaining reward.
I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our "creations," are simply the notes of our observations.
There is a world of created beings - living things, animals, entelechies, and souls - in the least part of matter.... Thus there is nothing waste, nothing sterile, nothing dead in the universe; no chaos, no confusions, save in appearance.
Capablanca plays very superficially sometimes, in a way that can only be ascribed to lack of concentration. This is an integral weakness of his make-up and can only be partially compensated by his employing his time allowance to the full.
What's the best part of being a mathematician? I'm not a religious man, but it's almost like being in touch with God when you're thinking about mathematics. God is keeping secrets from us, and it's fun to try to learn some of the secrets.
All models divide naturally...into two a priori parts: one is kinematics, whose aim is to parameterize the forms of the states of the process under consideration, and the other is dynamics, describing the evolution in time of these forms.
It may be said that the conceptions of differential quotient and integral, which in their origin certainly go back to Archimedes, were introduced into science by the investigations of Kepler, Descartes, Cavalieri, Fermat and Wallis. . . .
In mathematics, if a pattern occurs, we can go on to ask, Why does it occur? What does it signify? And we can find answers to these questions. In fact, for every pattern that appears, a mathematician feels he ought to know why it appears.
scientific thought does not mean thought about scientific subjects with long names. There are no scientific subjects. The subject of science is the human universe; that is to say, everything that is, or has been, or may be related to man.
A race preserves its vigor so long as it harbors a real contrast between what has been and what may be; and so long as it is nerved by the vigor to adventure beyond the safeties of the past. Without adventure civilization is in full decay.
I am giving this winter two courses of lectures to three students, of which one is only moderately prepared, the other less than moderately, and the third lacks both preparation and ability. Such are the onera of a mathematical profession.
The apex of mathematical achievement occurs when two or more fields which were thought to be entirely unrelated turn out to be closely intertwined. Mathematicians have never decided whether they should feel excited or upset by such events.