Quotes of All Topics . Occasions . Authors
Man is clearly made to think. It is his whole dignity and his whole merit; and his whole duty is to think as he ought. And the order of thought is to begin with ourselves, and with our Author and our end.
We know the truth not only through our reason but also through our heart. It is through the latter that we know first principles, and reason, which has nothing to do with it, tries in vain to refute them.
For as old age is that period of life most remote from infancy, who does not see that old age in this universal man ought not to be sought in the times nearest his birth, but in those most remote from it?
Mathematics, however, is, as it were, its own explanation; this, although it may seem hard to accept, is nevertheless true, for the recognition that a fact is so is the cause upon which we base the proof.
One cannot explain words without making incursions into the sciences themselves, as is evident from dictionaries; and, conversely, one cannot present a science without at the same time defining its terms.
One magnitude is said to be the limit of another magnitude when the second may approach the first within any given magnitude, however small, though the second may never exceed the magnitude it approaches.
In the realm of ideas, of mental objects, those ideas whose properties are reproducible are called mathematical objects, and the study of mental objects with reproducible properties is called mathematics.
At a purely formal level, one could call probability theory the study of measure spaces with total measure one, but that would be like calling number theory the study of strings of digits which terminate.
God is in the world, or nowhere, creating continually in us and around us. Insofar as man partakes of this creative process does he partake of the divine, of God, and that participation is his immortality.
Every science that has thriven has thriven upon its own symbols: logic, the only science which is admitted to have made no improvements in century after century, is the only one which has grown no symbols.
It was long before I got at the maxim, that in reading an old mathematician you will not read his riddle unless you plough with his heifer; you must see with his light, if you want to know how much he saw.
Some mathematicians didn't even perceive of the possibility of a picture being helpful. To the contrary, I went into an orgy of looking at pictures by the hundreds; the machines became a little bit better.
Everyone else would climb a peak by looking for a path somewhere in the mountain. Nash would climb another mountain altogether and from that distant peak would shine a searchlight back onto the first peak.
Those arts which are, to be sure, not finite, as geometry and arithmetic, do not suffer adornment; others, contrarily, are rather subject to division and embellishment, such as astronomy and jurisprudence.
Of what use would it be to you, sir, to become King of China on condition that you forgot what you have been? Would it not be the same as if God, at the same time he destroyed you, created a King in China?
You will die but the carbon will not; its career does not end with you. It will return to the soil, and there a plant may take it up again in time, sending it once more on a cycle of plant and animal life.
The success of the Allies in the west was in a measure offset by Teutonic victories in the east. When the invasion of Belgium began, Russia made immediate efforts to counteract by invasion of East Prussia.
He can be regarded as the great master of simplification. The art of resolving the tension at the critical moment and in the most effacious way so as to clarify the position as desired is Capablanca's own.
Science is concerned with what is possible while engineering is concerned with choosing, from among the many possible ways, one that meets a number of often poorly stated economic and practical objectives.
These ... tables (values of trignometric functions), constructed by means of new techniques based principally on the calculus of differences, are one of the most beautiful monuments ever erected to science.
There's also a sense of freedom. I was so obsessed by this problem that I was thinking about if all the time - when I woke up in the morning, when I went to sleep at night, and that went on for eight years.
It follows at once from the last proposition that the centre of gravity of any triangle is at the intersection of the lines drawn from any two angles to the middle points of the opposite sides respectively.
One couldn't even measure roughness. So, by luck, and by reward for persistence, I did found the theory of roughness, which certainly I didn't expect and expecting to found one would have been pure madness.
We conceal it from ourselves in vain - we must always love something. In those matters seemingly removed from love, the feeling is secretly to be found, and man cannot possibly live for a moment without it.
Each man is everything to himself, for with his death everything is dead for him. That is why each of us thinks he is everything to everyone. We must not judge nature by ourselves, but by its own standards.
The method of not erring is sought by all the world. The logicians profess to guide it, the geometricians alone attain it, and apart from science, and the imitations of it, there are no true demonstrations.
A few rules include all that is necessary for the perfection of the definitions, the axioms, and the demonstrations, and consequently of the entire method of the geometrical proofs of the art of persuading.
The further a mathematical theory is developed, the more harmoniously and uniformly does its construction proceed, and unsuspected relations are disclosed between hitherto separated branches of the science.
The tool which serves as intermediary between theory and practice, between thought and observation, is mathematics; it is mathematics which builds the linking bridges and gives the ever more reliable forms.
I propose to put forward an apology for mathematics; and I may be told that it needs none, since there are now few studies more generally recognized, for good reasons or bad, as profitable and praiseworthy.
They have the idea that non-commutative algebra should remind one of commutative algebra, but the former is more sophisticated. I believe that non-commutative algebra is just as simple, but it is different.
May not music be described as the mathematics of the sense, mathematics as music of the reason? The musician feels mathematics, the mathematician thinks music: music the dream, mathematics the working life.
The chief aim of all investigations of the external world should be to discover the rational order and harmony which has been imposed on it by God and which He revealed to us in the language of mathematics.
Definitions must contain the means of reaching a decision in a finite number off steps, and existence proofs must be conducted so that the quantity in question can be calculated with any degree of accuracy.
The Committee supports the idea that there should be, within the University of California, a campus which puts particular emphasis on the education of undergraduates within the framework of a College system.
Every new discovery in science brings with it a host of new problems, just as the invention of the automobile brought with it gas stations, roads, garages, mechanics, and a thousand other subsidiary details.
There is, however, another purpose to which academies contribute. When they consist of a limited number of persons, eminent for their knowledge, it becomes an object of ambition to be admitted on their list.
Mathematical discoveries, small or great are never born of spontaneous generation They always presuppose a soil seeded with preliminary knowledge and well prepared by labour, both conscious and subconscious.
One would have to have completely forgotten the history of science so as to not remember that the desire to know nature has had the most constant and the happiest influence on the development of mathematics.
I seem to be thinking rationally again in the style that is characteristic of scientists. However this is not entirely a matter of joy as if someone returned from physical disability to good physical health.
It would appear that we have reached the limits of what it is possible to achieve with computer technology, although one should be careful with such statements, as they tend to sound pretty silly in 5 years.
I see that the path of progress has never taken a straight line, but has always been a zigzag course amid the conflicting forces of right and wrong, truth and error, justice and injustice, cruelty and mercy.
I think schools generally do an effective and terribly damaging job of teaching children to be infantile, dependent, intellectually dishonest, passive and disrespectful to their own developmental capacities.
A mathematician experiments, amasses information, makes a conjecture, finds out that it does not work, gets confused and then tries to recover. A good mathematician eventually does so - and proves a theorem.
It is not possible to make a LASTING compromise between technology and freedom, because technology is by far the more powerful social force and continually encroaches on freedom through REPEATED compromises.
There is a technique, a knack, for thinking, just as there is for doing other things. You are not wholly at the mercy of your thoughts, any more than they are you. They are a machine you can learn to operate.
I was an undergraduate at Princeton, and I was pressed by the math department to go on to graduate school. Actually they gave me fellowships that paid my way, otherwise I would not have been able to continue.
Residues arise... naturally in several branches of analysis... Their consideration provides simple and easy-to-use methods, which are applicable to a large number of diverse questions, and some new results...
I spent half my life, roughly speaking, doing the study of nature in many aspects and half of my life studying completely artificial shapes. And the two are extraordinarily close; in one way both are fractal.
Father was bold, and Mother was cautious. They never shouted at each other but argued constantly about strategy, and they taught me very early that before taking big risks, one must carefully figure the odds.