Quotes of All Topics . Occasions . Authors
In formal logic, a contradiction is the signal of defeat, but in the evolution of real knowledge it marks the first step in progress toward a victory.
Religion is what the individual does with his own solitariness. Thus religion is solitariness; and if you are never solitary, you are never religious.
Why is geometry often described as cold and dry? One reason lies in its inability to describe the shape of a cloud, a mountain, a coastline or a tree.
The straight line has a property of self-similarity. Each piece of the straight line is the same as the whole line when used to a big or small extent.
Both chaos theory and fractal have had contacts in the past when they are both impossible to develop and in a certain sense not ready to be developed.
Most were beginning to feel they had learned enough to last for the rest of their lives. They remained mathematicians, but largely went their own way.
Love knows no limit to its endurance, no end to its trust, no fading of its hope; it can outlast anything. Love still stands when all else has fallen.
The Christian religion teaches me two points-that there is a God whom men can know, and that their nature is so corrupt that they are unworthy of Him.
As men are not able to fight against death, misery, ignorance, they have taken it into their heads, in order to be happy, not to think of them at all.
The finite is annihilated in the presence of the infinite, and becomes a pure nothing. So our spirit before God, so our justice before divine justice.
A mathematician ... has no material to work with but ideas, and so his patterns are likely to last longer, since ideas wear less with time than words.
I believe the geometric proportion served the creator as an idea when He introduced the continuous generation of similar objects from similar objects.
Mathematics is the surest way to immortality. If you make a big discovery in mathematics, you will be remembered after everyone else will be forgotten
Somebody came up to me after a talk I had given, and say, "You make mathematics seem like fun." I was inspired to reply, "If it isn't fun, why do it?"
Rather than pushing children to think like adults, we might do better to remember that they are great learners and to try harder to be more like them.
A lot of the math that I do, it's not sort of premeditated. I talk online or with a colleague, and I get interested, and I just follow where it leads.
It is the business of future to be dangerous.... The major advances in civilization are processes that all but wreck the societies in which they occur.
...the self-satisfied dogmatism with which mankind at each period of its history cherishes the delusion of the finality of existing modes of knowledge.
To be an abstraction does not mean that an entity is nothing. It merely means that its existence is only a factor of a more concrete element of nature.
No man ever believes with a true and saving faith unless God inclines his heart; and no man when God does incline his heart can refrain from believing.
Justice is what is established; and thus all our established laws will necessarily be regarded as just without examination, since they are established.
The parts of the universe ... all are connected with each other in such a way that I think it to be impossible to understand any one without the whole.
Nature has made all her truths independent of one another. Our art makes one dependent on the other. But this is not natural. Each keeps its own place.
Perhaps the most important principle on which the economy of a manufacture depends, is the division of labour amongst the persons who perform the work.
The primes are the raw material out of which we have to build arithmetic, and Euclid's theorem assures us that we have plenty of material for the task.
Absolute space, that is to say, the mark to which it would be necessary to refer the earth to know whether it really moves, has no objective existence.
All great scientists have, in a certain sense, been great artists; the man with no imagination may collect facts, but he cannot make great discoveries.
A function of a variable quantity is an analytic expression composed in any way whatsoever of the variable quantity and numbers or constant quantities.
Alekhine's real genius is in the preparation and construction of a position, long before combinations or mating attacks come into consideration at all.
For scholars and laymen alike it is not philosophy but active experience in mathematics itself that can alone answer the question: What is mathematics?
The result of teaching small parts of a large number of subjects is the passive reception of disconnected ideas, not illumed with any spark of vitality.
Then when I reached college I realized that many people had thought about the problem during the 18th and 19th centuries and so I studied those methods.
God only pours out his light into the mind after having subdued the rebellion of the will by an altogether heavenly gentleness which charms and wins it.
To be mistaken in believing that the Christian religion is true is no great loss to anyone; but how dreadful to be mistaken in believing it to be false!
All men naturally hate one another. I hold it a fact, that if men knew exactly what one says of the other, there would not be four friends in the world.
When a man who accepts the Christian doctrine lives unworthily of it, it is much clearer to say he is a bad Christian than to say he is not a Christian.
It is well known that the central problem of the whole of modern mathematics is the study of transcendental functions defined by differential equations.
A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.
... Philosophy proper is a subject, on the one hand so hopelessly obscure, on the other so astonishingly elementary, that there knowledge hardly counts.
The preoccupation with the choice of a mate both by male and female I regard as a continuing echo of the major selective force by which we have evolved.
In the early days of his reign, Bismarck confided to a friend that it would some day be necessary for Germany to confine William II in an insane asylum.
Newton took no exercise, indulged in no amusements, and worked incessantly, often spending eighteen or nineteen hours out of the twenty-four in writing.
I put forward as a general definition of civilization, that a civilized society is exhibiting the five qualities of Truth, Beauty, Adventure, Art, Peace.
A man really writes for an audience of about ten persons. Of course if others like it, that is clear gain. But if those ten are satisfied, he is content.
Then when I reached college, I realized that many people had thought about the problem during the 18th and 19th centuries and so I studied those methods.
Those who do not hate their own selfishness and regard themselves as more important than the rest of the world are blind because the truth lies elsewhere
The weather and my mood have little connection. I have my foggy and my fine days within me; my prosperity or misfortune has little to do with the matter.
I am coming more and more to the conviction that the necessity of our geometry cannot be demonstrated, at least neither by, nor for, the human intellect.
A person’s first duty, a young person’s at any rate, is to be ambitious, and the noblest ambition is that of leaving behind something of permanent value.
I wrote a great deal... but very little of any importance; there are not more than four of five papers which I can still remember with some satisfaction.