Quotes of All Topics . Occasions . Authors
In first place we must observe that the universe is spherical. This is either because that figure is the most perfect, as not being articulated, but whole and complete in itself; or because it is the most capacious and therefore best suited for that which is to contain and preserve all things.
I think the opportunity to deal with students and getting them properly oriented on science and theology and the relation between those is going to be important because science has been such an instrument used by the materialists to undermine the Christian faith and religious belief generally.
The aim of science is to seek the simplest explanation of complex facts. We are apt to fall into the error of thinking that the facts are simple because simplicity is the goal of our quest. The guiding motto in the life of every natural philosopher should be ``Seek simplicity and distrust it.''
[Christianity] endeavors equally to establish these two things: that God has set up in the Church visible signs to make himself known to those who should seek him sincerely, and that he has nevertheless so disguised them that he will only be perceived by those who seek him with all their heart.
Because of mathematics precise, formal character, mathematical arguments remain sound even when they are long and complex. In contast, common sense arguments can generally be trusted only if they remain short; even moderately long nonmathematical arguments rapidly becomes farfetched an dubious.
The scorn which I had reason to fear on account of the novelty and unconventionality of my opinion almost induced me to abandon completely the work which I had undertaken. . . . Astronomy is written for astronomers. To them my work too will seem, unless I am mistaken, to make some contribution.
Once in a while a person does only one thing in his whole life, and we'll talk about that later, but a lot of times there is repetition. I claim that luck will not cover everything. And I will cite Pasteur who said, "Luck favors the prepared mind." And I think that says it the way I believe it.
I tell adults about the experiences of more than a hundred teachers I've interviewed. They tell me that allowing the child to help them learn helped them become better teachers. That's because they no longer had to pretend they were the experts - not only about computers but about other things.
The Stoics say, "Retire within yourselves; it is there you will find your rest." And that is not true. Others say, "Go out of yourselves; seek happiness in amusement." And this is not true. Illness comes. Happiness is neither without us nor within us. It is in God, both without us and within us.
He will also find that the high and independent spirit, which usually dwells in the breast of those who are deeply versed in scientific pursuits, is ill adapted for administrative appointments; and that even if successful, he must hear many things he disapproves, and raise no voice against them.
Mechanical Notation ... I look upon it as one of the most important additions I have made to human knowledge. It has placed the construction of machinery in the rank of a demonstrative science. The day will arrive when no school of mechanical drawing will be thought complete without teaching it.
Besides it is an error to believe that rigour is the enemy of simplicity. On the contrary we find it confirmed by numerous examples that the rigorous method is at the same time the simpler and the more easily comprehended. The very effort for rigor forces us to find out simpler methods of proof.
There is always more in one of Ramanujan's formulae than meets the eye, as anyone who sets to work to verify those which look the easiest will soon discover. In some the interest lies very deep, in others comparatively near the surface; but there is not one which is not curious and entertaining.
Science is composed of laws which were originally based on a small, carefully selected set of observations, often not very accurately measured originally; but the laws have later been found to apply over much wider ranges of observations and much more accurately than the original data justified.
If you want to think new thoughts that are different, then do what creative people do - get the problem reasonably clear and then refuse to look at any answers until you've thought the problem through carefully how you would do it, how you could slightly change the problem to be the correct one.
The oneness of the universe, and the oneness of each element of the universe, repeat themselves to the crack of doom in the creative advance from creature to creature, each creature including in itself the whole of history and exemplifying the self-identity of things and their mutual diversities.
Having visual impressions is, of course, necessary for seeing things, but it is not sufficient. What must be added is not anything sensible. And it is precisely this that unlocks the outer world for us; for without this non-sensible something, each of us would remain locked up in his inner world.
Astronomy is useful because it raises us above ourselves; it is useful because it is grand; .... It shows us how small is man's body, how great his mind, since his intelligence can embrace the whole of this dazzling immensity, where his body is only an obscure point, and enjoy its silent harmony.
Other nations of different habits are not enemies: they are godsends. Men require of their neighbours something sufficiently akin to be understood, something sufficiently different to provoke attention, and something great enough to command admiration. We must not expect, however, all the virtues.
There are problems to whose solution I would attach an infinitely greater importance than to those of mathematics, for example touching ethics, or our relation to God, or concerning our destiny and our future; but their solution lies wholly beyond us and completely outside the province of science.
Mathematics is the most exact science, and its conclusions are capable of absolute proof. But this is so only because mathematics does not attempt to draw absolute conclusions. All mathematical truths are relative, conditional. In E. T. Bell Men of Mathematics, New York: Simona and Schuster, 1937.
There is no ground for supposing that matter and energy existed before [the Big Bang] and were suddenly galvanized into action. For what could distinguish that moment from all other moments in eternity? It is simpler to postulate creation ex nihilo-Divine will constituting Nature from nothingness.
Mathematicians also make terrible salesmen. Physicists can discover the same thing as a mathematician and say 'We've discovered a great new law of nature. Give us a billion dollars.' And if it doesn't change the world, then they say, 'There's an even deeper thing. Give us another billion dollars.'
Yet in this my stars were not Mercury as morning star in the angle of the seventh house, in quartile with Mars, but they were Copernicus, they were Tycho Brahe, without whose books of observations everything which has now been brought by me into the brightest daylight would lie buried in darkness.
Reality cannot be found except in One single source, because of the interconnection of all things with one another. I maintain also that substances, whether material or immaterial, cannot be conceived in their bare essence without any activity, activity being of the essence of substance in general.
Virtue is not a mushroom, that springeth up of itself in one night when we are asleep, or regard it not; but a delicate plant, that groweth slowly and tenderly, needing much pains to cultivate it, much care to guard it, much time to mature it, in our untoward soil, in this world's unkindly weather.
It is true that I should have been surprised in the past to learn that Professor Hardy had joined the Oxford Group. But one could not say the adverse chance was 1:10. Mathematics is a dangerous profession; an appreciable proportion of us go mad, and then this particular event would be quite likely.
In the realm of science, all attempts to find any evidence of supernatural beings, of metaphysical concepts, as God, immortality, infinity, etc have thus far failed, and if we are honest, we must confess that in science there exists no God, no immortality, no soul or mind, as distinct from the body.
Mass, time , magnetic moment, the unconscious: we have grown up with these symbolic concepts, so that we are startled to be told that man had once to create them for himself. He had indeed, and he has: for mass is not an intuition in the muscle, and time is not bought ready-made at the watchmaker's.
Nature is more subtle, more deeply intertwined and more strangely integrated than any of our pictures of her than any of our errors. It is not merely that our pictures are not full enough; each of our pictures in the end turns out to be so basically mistaken that the marvel is that it worked at all.
Napoleon: You have written this huge book on the system of the world without once mentioning the author of the universe. Laplace: Sire, I had no need of that hypothesis. Later when told by Napoleon about the incident, Lagrange commented: Ah, but that is a fine hypothesis. It explains so many things.
When a truth is necessary, the reason for it can be found by analysis, that is, by resolving it into simpler ideas and truths until the primary ones are reached. It is this way that in mathematics speculative theorems and practical canons are reduced by analysis to definitions, axioms and postulates.
So in the end it wasn't Gödel, it wasn't Turing, and it wasn't my results that are making mathematics go into an experimental mathematics direction, in a quasi-empirical direction. The reason why mathematicians are changing their working habits is the computer. I think that this is an excellent joke!
When I think about my own work, there usually is one or the other light bulb moment where there is this flash of insight. Where do these ideas come from? Supernatural? Enigmatic? Divine? You can take that any way you like. I think the human mind does a lot of things that we could call somehow divine.
Perhaps there will be prattlers who, although completely ignorant of mathematics, nevertheless take it upon themselves to pass judgment on mathematical questions, and on account of some passage in Scripture, badly distorted to their purpose, will dare to censure and assail what I have presented here.
Long intervals frequently elapse between the discovery of new principles in science and their practical application... Those intellectual qualifications, which give birth to new principles or to new methods, are of quite a different order from those which are necessary for their practical application.
If we ought not to fear mortal truth, still less should we dread scientific truth. In the first place it can not conflict with ethics? But if science is feared, it is above all because it can give no happiness? Man, then, can not be happy through science but today he can much less be happy without it.
Every phenomenon, however trifling it be, has a cause, and a mind infinitely powerful, and infinitely well-informed concerning the laws of nature could have foreseen it from the beginning of the ages. If a being with such a mind existed, we could play no game of chance with him; we should always lose.
To be sure, mathematics can be extended to any branch of knowledge, including economics, provided the concepts are so clearly defined as to permit accurate symbolic representation. That is only another way of saying that in some branches of discourse it is desirable to know what you are talking about.
We believe that arithmetic as it has been taught in grade schools until quite recently has such a meagre intellectual content that the oft-noted reaction against the subject is not an unfortunate rebellion against a difficult subject, but a perfectly proper response to a preoccupation with triviality.
If you attack a mathematical problem directly, very often you come to a dead end, nothing you do seems to work and you feel that if only you could peer round the corner there might be an easy solution. There is nothing like having somebody else beside you, because he can usually peer round the corner.
'Facts, facts, facts,' cries the scientist if he wants to emphasize the necessity of a firm foundation for science. What is a fact? A fact is a thought that is true. But the scientist will surely not recognize something which depends on men's varying states of mind to be the firm foundation of science.
Great is God our Lord, great is His power and there is no end to His wisdom. Praise Him you heavens, glorify Him, sun and moon and you planets. For out of Him and through Him, and in Him are all things..... We know, oh, so little. To Him be the praise, the honor and the glory from eternity to eternity.
I loved doing problems in school. I'd take them home and make up new ones of my own. But the best problem I ever found, I found in my local public library. I was just browsing through the section of math books and I found this one book, which was all about one particular problem - Fermat's Last Theorem.
I am persuaded that this method [for calculating the volume of a sphere] will be of no little service to mathematics. For I foresee that once it is understood and established, it will be used to discover other theorems which have not yet occurred to me, by other mathematicians, now living or yet unborn.
When I consider the small span of my life absorbed in the eternity of all time, or the small part of space which I can touch or see engulfed by the infinite immensity of spaces that I know not and that know me not, I am frightened and astonished to see myself here instead of there … now instead of then.
I have seen something of the project of M. de St. Pierre, for maintaining a perpetual peace in Europe. I am reminded of a device in a cemetery, with the words: Pax perpetua ; for the dead do not fight any longer: but the living are of another humor; and the most powerful do not respect tribunals at all.
I have said that science is impossible without faith. ... Inductive logic, the logic of Bacon, is rather something on which we can act than something which we can prove, and to act on it is a supreme assertion of faith ... Science is a way of life which can only fluorish when men are free to have faith.
Most remarks made by children consist of correct ideas very badly expressed. A good teacher will be very wary of saying 'No, that's wrong.' Rather, he will try to discover the correct idea behind the inadequate expression. This is one of the most important principles in the whole of the art of teaching.
Let us imagine a number of men in chains and all condemned to death, where some are killed each day in the sight of the others, and those who remain see their own fate in that of their fellows and wait their turn, looking at each other sorrowfully and without hope. It is an image of the condition of man.