Quotes of All Topics . Occasions . Authors
What use is it to us to hear it said of a man that he has thrown off the yoke that he does not believe there is a God to watch over his actions, that he reckons himself the sole master of his behavior, and that he does not intend to give an account of it to anyone but himself?
On Sept 15th [1852] Mr Goulburn, Chancellor of the Exchequer, asked my opinion on the utility of Mr Babbage's calculating machine, and the propriety of spending further sums of money on it. I replied, entering fully into the matter, and giving my opinion that it was worthless.
The sphere of the attractive virtue which is in the moon extends as far as the earth, and entices up the waters; but as the moon flies rapidly across the zenith, and the waters cannot follow so quickly, a flow of the ocean is occasioned in the torrid zone towards the westward.
If Baltimore's view, that scientists who do not take the words of authorities are far removed from the ordinary behavior of scientists, prevails in the scientific community, then something fundamental, very serious, and very disturbing is happening to the scientific community.
As one reads mathematics, one needs to have an active mind, asking questions, forming mental connections between the current topic and other ideas from other contexts, so as to develop a sense of the structure, not just familiarity with a particular tour through the structure.
Since [man] is infinitely removed from comprehending the extremes, the end of things and their beginning are hopelessly hidden from him in an impenetrable secret; he is equally incapable of seeing the nothing from which he was made, and the infinite in which he is swallowed up.
I don't have any particular recipe. It is the reason why doing research is challenging as well as attractive. It is like being lost in a jungle and trying to use all the knowledge that you can gather to come up with some new tricks, and with some luck, you might find a way out.
When the weather changes and hurricanes hit, nobody believes that the laws of physics have changed. Similarly, I don't believe that when the stock market goes into terrible gyrations its rules have changed. It's the same stock market with the same mechanisms and the same people.
Without deductive logic science would be entirely useless. It is merely a barren game to ascend from the particular to the general, unless afterwards we can reverse the process and descend from the general to the particular, ascending and descending like angels on Jacob's ladder.
Everybody in mathematics had given up for 100 years or 200 years the idea that you could from pictures, from looking at pictures, find new ideas. That was the case long ago in the Middle Ages, in the Renaissance, in later periods, but then mathematicians had become very abstract.
Therefore, those to whom God has imparted religion by intuition are very fortunate and justly convinced. But to those who do not have it, we can give it only by reasoning, waiting for God to give them spiritual insight, without which faith is only human and useless for salvation.
We lay down a fundamental principle of generalization by abstraction: The existence of analogies between central features of various theories implies the existence of a general theory which underlies the particular theories and unifies them with respect to those central features.
One is hard pressed to think of universal customs that man has successfully established on earth. There is one, however, of which he can boast the universal adoption of the Hindu-Arabic numerals to record numbers. In this we perhaps have man's unique worldwide victory of an idea.
[Mathematics] unceasingly calls forth the faculties of observation and comparison; one of its principal weapons is induction: it has frequent recourse to trial and verification; and it affords a boundless scope for the exercise of the highest efforts of imagination and invention.
Therefore, the central point which we see in the centre of the hieroglyphic Monad produces the Earth , round which the Sun , the Moon , and the other planets follow their respective paths. The Sun has the supreme dignity , and we represent him by a circle having a visible centre.
I shall devote all my efforts to bring light into the immense obscurity that today reigns in Analysis. It so lacks any plan or system, that one is really astonished that there are so many people who devote themselves to it - and, still worse, it is absolutely devoid of any rigor.
The simplicity of nature is not to be measured by that of our conceptions. Infinitely varied in its effects, nature is simple only in its causes, and its economy consists in producing a great number of phenomena, often very complicated, by means of a small number of general laws.
The ideas of Freud were popularized by people who only imperfectly understood them, who were incapable of the great effort required to grasp them in their relationship to larger truths, and who therefore assigned to them a prominence out of all proportion to their true importance.
I would inquire of reasonable persons whether this principle: Matter is naturally wholly incapable of thought, and this other: I think, therefore I am, are in fact the same in the mind of Descartes, and in that of St. Augustine, who said the same thing twelve hundred years before.
Neumann, to a physicist seeking help with a difficult problem: Simple. This can be solved by using the method of characteristics. Physicist: I'm afraid I don't understand the method of characteristics. Neumann: In mathematics you don't understand things. You just get used to them.
With the exception of the geometrical series, there does not exist in all of mathematics a single infinite series the sum of which has been rigorously determined. In other words, the things which are the most important in mathematics are also those which have the least foundation.
We are in the position of the man who has only two ambitions in life. One is to invent the universal solvent which will dissolve any solid substance, and the second is to invent the universal container which will hold any liquid. Whatever this inventor does, he will be frustrated.
Many teachers are concerned about the amount of material they must cover in a course. One cynic suggested a formula: since, he said, students on the average remember only about 40% of what you tell them, the thing to do is to cram into each course 250% of what you hope will stick.
Man is nothing but insincerity, falsehood, and hypocrisy, both in regard to himself and in regard to others. He does not wish that he should be told the truth, he shuns saying it to others; and all these moods, so inconsistent with justice and reason, have their roots in his heart.
Finally, two days ago, I succeeded - not on account of my hard efforts, but by the grace of the Lord. Like a sudden flash of lightning, the riddle was solved. I am unable to say what was the conducting thread that connected what I previously knew with what made my success possible.
It can happen to but few philosophers, and but at distant intervals, to snatch a science, like Dalton, from the chaos of indefinite combination, and binding it in the chains of number, to exalt it to rank amongst the exact. Triumphs like these are necessarily 'few and far between.'
Trimming consists of clipping off little bits here and there from those observations which differ most in excess from the mean, and in sticking them onto those which are too small; a species of 'equitable adjustment,' as a radical would term it, which cannot be admitted in science.
If two stones were placed... near each other, and beyond the sphere of influence of a third cognate body, these stones, like two magnetic needles, would come together in the intermediate point, each approaching the other by a space proportional to the comparative mass of the other.
I think we should allow for schools within schools, where 100 out of 500 kids may be organized by the way they work and what they do, and what they do often is more progressive. I would like to see a lot of kids of different ages, maybe even some adults, work together on a project.
Undoubtedly equality of goods is just; but, being unable to cause might to obey justice, men has made it just to obey might. Unable to strengthen justice, they have justified might--so that the just and the strong should unite, and there should be peace, which is the sovereign good.
...great difficulties are felt at first and these cannot be overcome except by starting from experiments .. and then be conceiving certain hypotheses ... But even so, very much hard work remains to be done and one needs not only great perspicacity but often a degree of good fortune.
People tend to think that mathematicians always work in sterile conditions, sitting around and staring at the screen of a computer, or at a ceiling, in a pristine office. But in fact, some of the best ideas come when you least expect them, possibly through annoying industrial noise.
Since we tend to see ourselves primarily in the light of our intentions, which are invisible to others, while we see others mainly in the light of their actions, which are all that's visible to us, we have a situation in which misunderstanding and injustice are the order of the day.
Kurt Godel's achievement in modern logic is singular and monumental - indeed it is more than a monument, it is a landmark which will remain visible far in space and time. ... The subject of logic has certainly completely changed its nature and possibilities with Godel's achievement.
Who would talk about the evolutionary mind? Who cares about the good and evil in the evolution of species, and so on? This must be interesting only to the degree to which it informs us in this very present moment regarding our choices that we will make in the creation of our future.
There may be such a thing as habitual luck. People who are said to be lucky at cards probably have certain hidden talents for those games in which skill plays a role. It is like hidden parameters in physics, this ability that does not surface and that I like to call "habitual luck".
I am sure that one secret of a successful teacher is that he has formulated quite clearly in his mind what the pupil has got to know in precise fashion. He will then cease from half-hearted attempts to worry his pupils with memorizing a lot of irrelevant stuff of inferior importance.
Parents fear the destruction of natural affection in their children. What is this natural principle so liable to decay? Habit is a second nature, which destroys the first. Why is not custom nature? I suspect that this nature itself is but a first custom, as custom is a second nature.
Where should I start? Start from the statement of the problem. ... What can I do? Visualize the problem as a whole as clearly and as vividly as you can. ... What can I gain by doing so? You should understand the problem, familiarize yourself with it, impress its purpose on your mind.
There do remain dispersed in the soil of human nature divers seeds of goodness, of benignity, of ingenuity, which, being cherished, excited, and quickened by good culture, do, by common experience, thrust out flowers very lovely, and yield fruits very pleasant of virtue and goodness.
Gradually I began to intellectually reject some of the delusionally influenced lines of thinking which had been characteristic of my orientation. This began, most recognizably, with the rejection of politically-oriented thinking as essentially a hopeless waste of intellectual effort.
To know the mighty works of God, to comprehend His wisdom and majesty and power; to appreciate, in degree, the wonderful workings of His laws, surely all this must be a pleasing and acceptable mode of worship to the Most High, to whom ignorance cannot be more grateful than knowledge.
By an incredible coincidence, Gamow and Edward Condon, who had discovered simultaneously and independently the explanation of radioactivity (one in Russia, the other in this country), came to spend the the last ten years of their lives within a hundred yards of each other in Boulder.
Pride counterbalances all our miseries, for it either hides them, or, if it discloses them, boasts of that disclosure. Pride has such a thorough possession of us, even in the midst of our miseries and faults, that we are prepared to sacrifice life with joy, if it may but be talked of.
Precedents are treated by powerful minds as fetters with which to bind down the weak, as reasons with which to mistify the moderately informed, and as reeds which they themselves fearlessly break through whenever new combinations and difficult emergencies demand their highest efforts.
I do not see that the sex of the candidate is an argument against her admission as a Privatdozent. After all, the Senate is not a bathhouse. Objecting to sex discrimination being the reason for rejection of Emmy Noether's application to join the faculty at the University of Gottingen.
What is a good definition? For the philosopher or the scientist, it is a definition which applies to all the objects to be defined, and applies only to them; it is that which satisfies the rules of logic. But in education it is not that; it is one that can be understood by the pupils.
Often when works at a hard question, nothing good is accomplished at the first attack. Then one takes a rest, long or short, and sits down anew to the work. During the first half-hour, as before, nothing is found, and then all of a sudden the decisive idea presents itself to the mind.
The fruits of the earth do not more obviously require labor and cultivation to prepare them for our use and subsistence, than our faculties demand instruction and regulation in order to qualify us to become upright and valuable members of society, useful to others, or happy ourselves.
Given a conjecture, the best thing is to prove it. The second best thing is to disprove it. The third best thing is to prove that it is not possible to disprove it, since it will tell you not to waste your time trying to disprove it. That's what Gödel did for the Continuum Hypothesis.