Quotes of All Topics . Occasions . Authors
The mentality of mankind and the language of mankind created each other. If we like to assume the rise of language as a given fact, then it is not going too far to say that the souls of men are the gift from language to mankind. The account of the sixth day should be written: He gave them speech, and they became souls.
There are two excesses: to exclude reason, to admit nothing but reason. The supreme achievement of reason is to realise that there is a limit to reason. Reason's last step is the recognition that there are an infinite number of things which are beyond it. It is merely feeble if it does not go as far as to realise that.
When the logician has resolved each demonstration into a host of elementary operations, all of them correct, he will not yet be in possession of the whole reality, that indefinable something that constitutes the unity ... Now pure logic cannot give us this view of the whole; it is to intuition that we must look for it.
Geometry, which before the origin of things was coeternal with the divine mind and is God himself (for what could there be in God which would not be God himself?), supplied God with patterns for the creation of the world, and passed over to Man along with the image of God; and was not in fact taken in through the eyes.
You can sit down with your child and prompt him to show you something - perhaps how to play a game [on the computer]. By learning a game, you're getting close to the kid and gaining insight into ways of learning. The kid can see this happening and feels respected, so it fosters the relationship between you and the kid.
If we dreamed the same thing every night, it would affect us much as the objects we see every day. And if a common workman were sure to dream every night for twelve hours that he was a king, I believe he would be almost as happy as a king who should dream every night for twelve hours on end that he was a common workman.
Unless there exist peculiar institutions for the support of such inquirers, or unless the Government directly interfere, the contriver of a thaumatrope may derive profit from his ingenuity, whilst he who unravels the laws of light and vision, on which multitudes of phenomena depend, shall descend unrewarded to the tomb.
Mathematics has two faces: it is the rigorous science of Euclid, but it is also something else. Mathematics presented in the Euclidean way appears as a systematic, deductive science; but mathematics in the making appears as an experimental, inductive science. Both aspects are as old as the science of mathematics itself.
Mathematicians have never been in full agreement on their science, though it is said to be the science of self-evident verities -- absolute, indisputable and definitive. They have always been in controversy over developing aspects of mathematics, and they have always considered their own age to be in a period of crisis.
Why waste words? Geometry existed before the Creation, is co-eternal with the mind of God, is God himself (what exists in God that is not God himself?): geometry provided God with a model for the Creation and was implanted into man, together with God's own likeness - and not merely conveyed to his mind through the eyes.
Nature gets credit which should in truth be reserved for ourselves: the rose for its scent, the nightingale for its song; and the sun for its radiance. The poets are entirely mistaken. They should address their lyrics to themselves and should turn them into odes of self congratulation on the excellence of the human mind.
The full impact of the Lobachevskian method of challenging axioms has probably yet to be felt. It is no exaggeration to call Lobachevsky the Copernicus of Geometry [as did Clifford], for geometry is only a part of the vaster domain which he renovated; it might even be just to designate him as a Copernicus of all thought.
Surely with as good reason as had Archimedes to have the cylinder, cone and sphere engraved on his tombstone might our distinguished countrymen leave testamentary directions for the cubic eikosiheptagram to be engraved on theirs. Spirit of the Universe! wither are we drifting, and when, where, and how is all this to end?
In attempting to construct such (artificially intelligent) machines we should not be irreverently usurping His (God's) power of creating souls, any more than we are in the procreation of children,” Turing had advised. “Rather we are, in either case, instruments of His will providing mansions for the souls that He creates.
It is certain that those who have the living faith in their hearts see at once that all existence is none other than the work of the God whom they adore. But for those in whom this light is extinguished, [if we were to show them our proofs of the existence of God] nothing is more calculated to arouse their contempt. . . .
All the dignity of man consists in thought. Thought is therefore by its nature a wonderful and incomparable thing. It must have strange defects to be contemptible. But it has such, so that nothing is more ridiculous. How great it is in its nature! How vile it is in its defects! But what is this thought? How foolish it is!
Mathematics is not a contemplative but a creative subject; no one can draw much consolation from it when he has lost the power or the desire to create; and that is apt to happen to a mathematician rather soon. It is a pity, but in that case he does not matter a great deal anyhow, and it would be silly to bother about him.
The first and foremost duty of the high school in teaching mathematics is to emphasize methodical work in problem solving...The teacher who wishes to serve equally all his students, future users and nonusers of mathematics, should teach problem solving so that it is about one-third mathematics and two-thirds common sense.
A great discovery solves a great problem, but there is a grain of discovery in the solution of any problem. Your problem may be modest, but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery.
It is possible, of course, to operate with figures mechanically, just as it is possible to speak like a parrot: but that hardly deserves the names of thought. It only becomes possible at all after the mathematical notation has, as a result of genuine thought, been so developed that it does the thinking for us, so to speak.
Tolstoi explains somewhere in his writings why, in his opinion, “Science for Science's sake” is an absurd conception. We cannot know all the facts, since they are practically infinite in number. We must make a selection. Is it not better to be guided by utility, by our practical, and more especially our moral, necessities?
The discoveries of science, the works of art are explorations - more, are explosions, of a hidden likeness. The discoverer or artist presents in them two aspects of nature and fuses them into one. This is the act of creation, in which an original thought is born, and it is the same act in original science and original art.
When the most abstract and "useless" disciplines have been cultivated for a time, they are often seized upon as practical tools by other departments of science. I conceive that this is no accident, as if one bought a top hat for a wedding, and discovered later when a fire broke out, that it could be used as a water bucket.
Mathematics is a form of poetry which transcends poetry in that it proclaims a truth; a form of reasoning which transcends reasoning in that it wants to bring about the truth it proclaims; a form of action, of ritual behavior, which does not find fulfilment in the act but must proclaim and elaborate a poetic form of truth.
I do not remember having felt, as a boy, any passion for mathematics, and such notions as I may have had of the career of a mathematician were far from noble. I thought of mathematics in terms of examinations and scholarships: I wanted to beat other boys, and this seemed to be the way in which I could do so most decisively.
[Regarding mathematics,] there are now few studies more generally recognized, for good reasons or bad, as profitable and praiseworthy. This may be true; indeed it is probable, since the sensational triumphs of Einstein, that stellar astronomy and atomic physics are the only sciences which stand higher in popular estimation.
Neither the circle without the line, nor the line without the point, can be artificially produced. It is, therefore, by virtue of the point and the Monad that all things commence to emerge in principle. That which is affected at the periphery, however large it may be, cannot in any way lack the support of the central point.
The extraordinary fact is that the first idea I had which motivated me, that worked, is conjecture, a mathematical idea which may or may not be true. And that idea is still unproven. It is the foundation, what started me and what everybody failed to **** prove has so far defeated the greatest efforts by experts to be proven.
People feel that the Bible is unequivocal in stating that the age of the earth is very young and so on and so forth, and so the big things get lumped together with the lesser things. And the age of the earth is for example virtually made a touchstone of doctrine, when there's so much evidence out there in science against it.
You are in the same manner surrounded with a small circle of persons... full of desire. They demand of you the benefits of desire... You are therefore properly the king of desire. ...equal in this to the greatest kings of the earth... It is desire that constitutes their power; that is, the possession of things that men covet.
We do not rest satisfied with the present.... So imprudent we are that we wander in the times which are not ours and do not thinkof the only one which belongs to us; and so idle are we that we dream of those times which are no more and thoughtlessly overlook that which alone exists. For the present is generally painful to us.
To teach effectively a teacher must develop a feeling for his subject; he cannot make his students sense its vitality if he does not sense it himself. He cannot share his enthusiasm when he has no enthusiasm to share. How he makes his point may be as important as the point he makes; he must personally feel it to be important.
In the broad light of day mathematicians check their equations and their proofs, leaving no stone unturned in their search for rigour. But, at night, under the full moon, they dream, they float among the stars and wonder at the miracle of the heavens. They are inspired. Without dreams there is no art, no mathematics, no life.
Regular geometry, the geometry of Euclid, is concerned with shapes which are smooth, except perhaps for corners and lines, special lines which are singularities, but some shapes in nature are so complicated that they are equally complicated at the big scale and come closer and closer and they don't become any less complicated.
There is a lot of difference between tempting and leading into error. God tempts but does not lead into error. To tempt is to provide opportunities for us to do certain things if we do not love God, but putting us under no necessity to do so. To lead into error is to compel a man necessarily to conclude and follow a falsehood.
All the scientist creates in a fact is the language in which he enunciates it. If he predicts a fact, he will employ this language, and for all those who can speak and understand it, his prediction is free from ambiguity. Moreover, this prediction once made, it evidently does not depend upon him whether it is fulfilled or not.
The idea of separating church and state by the Founding Fathers of America was freedom from the domination of one form of religion, because many of them left England, because they were persecuted by the church, because they want to express their Christian faith in a different way. So it was a bit of warfare between Christians.
The subject for which I am asking your attention deals with the foundations of mathematics. To understand the development of the opposing theories existing in this field one must first gain a clear understnding of the concept "science"; for it is as a part of science that mathematics originally took its place in human thought.
Educators who have said, "We don't like that, so we'll continue to teach as if it's not happening," are just aggravating the gap between what happens in schools and what happens in the real world. Because of their personalities, or for cultural reasons, some kids might better express themselves through moving images and sound.
The progress of Science consists in observing interconnections and in showing with a patient ingenuity that the events of this ever-shifting world are but examples of a few general relations, called laws. To see what is general in what is particular, and what is permanent in what is transitory, is the aim of scientific thought.
We know then the existence and nature of the finite, because we also are finite and have extension. We know the existence of the infinite and are ignorant of its nature, because it has extension like us, but not limits like us. But we know neither the existence nor the nature of God, because he has neither extension nor limits.
I'm not the sort of person who does my mathematics writing on paper. I do that at the last stage of the game. I do my mathematics in my head. I sit down for a hard day's work and I write nothing all day. I just think. And I walk up and down because that helps keep me awake, it keeps the blood circulating, and I think and think.
A different image came to me a few weeks ago. The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration... the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it... yet finally it surrounds the resistant substance.
The existence of these patterns [fractals] challenges us to study forms that Euclid leaves aside as being formless, to investigate the morphology of the amorphous. Mathematicians have disdained this challenge, however, and have increasingly chosen to flee from nature by devising theories unrelated to anything we can see or feel.
Since we cannot be universal and know all that is to be known of everything, we ought to know a little about everything. For it is far better to know something about everything than to know all about one thing. This universality is the best. If we can have both, still better; but if we must choose, we ought to choose the former.
Medals are great encouragement to young men and lead them to feel their work is of value, I remember how keenly I felt this when in the 1890s. I received the Darwin Medal and the Huxley Medal. When one is old, one wants no encouragement and one goes on with one's work to the extent of one's power, because it has become habitual.
The new book by Nafeez Ahmed, based on very extensive and deep research, is by far the best on the 9/11 syndrome. Articulating and documenting what many feel, and empowering them into action, the book will have an impact on entrenched US empire elites unwilling and unable to take it on. Votes of thanks to Ahmed, and to Interlink!
If there should chance to be any mathematicians who, ignorant in mathematics yet pretending to skill in that science, should dare, upon the authority of some passage of Scripture wrested to their purpose, to condemn and censure my hypothesis, I value them not, and scorn their inconsiderate judgement. De Revolutionibus Coelestibus
Biot, who assisted Laplace in revising it [The Mécanique Céleste] for the press, says that Laplace himself was frequently unable to recover the details in the chain of reasoning, and if satisfied that the conclusions were correct, he was content to insert the constantly recurring formula, 'Il est àisé a voir' [it is easy to see].
You have to understand, in the current academic climate, Intelligent Design is like leprosy or heresy in times past. To be tagged as an ID supporter is to become an academic pariah, and this holds even at so-called Christian institutions that place a premium on respectability at the expense of truth and the offense of the Gospel.