Infinity Quotes - page 13
The writer has developed a "dynamical” approach to the study of cooperative games based upon reduction to non-cooperative form. One proceeds by constructing a model of the preplay negotiation so that the steps of negotiation become moves in a larger non-cooperative game [which will have an infinity of pure strategies] describing the total situation. This larger game is then treated in terms of the theory of this paper [extended to infinite games] and if values are obtained they are taken as the values of the cooperative game. Thus the problem of analyzing a cooperative game becomes the problem of obtaining a suitable, and convincing, non-cooperative model for the negotiation.
The writer has, by such a treatment, obtained values for all finite two-person cooperative games, and some special n-person games.
John Forbes Nash
A great profusion of things, which are splendid or valuable in themselves, is magnificent. The starry heaven, though it occurs so very frequently to our view, never fails to excite an idea of grandeur. This cannot be owing to the stars themselves, separately considered. The number is certainly the cause. The apparent disorder augments the grandeur, for the appearance of care is highly contrary to our idea of magnificence. Besides, the stars lie in such apparent confusion, as makes it impossible on ordinary occasions to reckon them. This gives them the advantage of a sort of infinity.
Edmund Burke
What it is, only God knows. In terms of matter I suppose the thing Ammi described would be called a gas, but this gas obeyed the laws that are not of our cosmos. This was no fruit of such worlds and suns as shine on the telescopes and photographic plates of our observatories. This was no breath from the skies whose motions and dimensions our astronomers measure or deem too vast to measure. It was just a colour out of space - a frightful messenger from unformed realms of infinity beyond all Nature as we know it; from realms whose mere existence stuns the brain and numbs us with the black extra-cosmic gulfs it throws open before our frenzied eyes.
H. P. Lovecraft
These primitive propositions ... suffice to deduce all the properties of the numbers that we shall meet in the sequel. There is, however, an infinity of systems which satisfy the five primitive propositions. ... All systems which satisfy the five primitive propositions are in one-to-one correspondence with the natural numbers. The natural numbers are what one obtains by abstraction from all these systems; in other words, the natural numbers are the system which has all the properties and only those properties listed in the five primitive propositions.
Giuseppe Peano
A tiny wren perched on top of a bramble not ten feet from him and trilled its violent song. He saw its glittering black eyes, the red and yellow of its song-gaped throat - a midget ball of feathers that yet managed to make itself the Announcing Angel of evolution: I am that I am, thou shalt not pass my being now. He stood as Pisanello's saint stood, astonished perhaps more at his own astonishment at this world's existing so close, so within reach of all that suffocating banality of ordinary day. In those few moments of defiant song, any ordinary hour or place - and therefore the vast infinity of all Charles's previous hours and places - seemed vulgarized, coarsened, made garish. The appallimg ennui of human reality lay cleft to the core; and the heart of all life pulsed there in the wren's triumphant throat.
John Fowles
There cannot be a single, simple body which is infinite, either, as some hold, one distinct from the elements, which they then derive from it, nor without this qualification. For there are some who make this (i.e. a body distinct from the elements) the infinite, and not air or water, in order that the other things may not be destroyed by their infinity. They are in opposition one to another - air is cold, water moist, and fire hot-and therefore, if any one of them were infinite, the rest would have ceased to be by this time. Accordingly they say that what is infinite is something other than the elements, and from it the elements arise.
Anaximander
There is a fifth dimension beyond that which is known to man. It is a dimension as vast as space and as timeless as infinity. It is the middle ground between light and shadow, between science and superstition, and it lies between the pit of man's fears and the summit of his knowledge. This is the dimension of imagination. It is an area which we call the Twilight Zone.
Rod Serling
We had an infinity of time: The Honorable Elijah Muhammad says that God taught him over 78 trillion years of time on our planet-not his (the White Man.) Our planet, in Our universe-we are the makers of all that you see of creation. And when a son would be born to end the rule of the enemy of God, his (the enemy's) job would have been made complete. Now, you can be angry with White people if you want to, and you could curse them out and say ugly things, because they have done ugly things. But wait a minute: They couldn't have done any of this if Allah (God) didn't permit it. He (the White Man) justifies his existence as a "devil” by being himself.
Louis Farrakhan
Only those with their eyes open to the world of nature are capable of uncovering its truth. Everything springs from a sense of gratitude toward nature. Aikido, though praised as a healthful system of self-defense techniques, would be nothing apart from the laws of the great universe. The martial way begins and ends with courtesy, itself an attitude of thankfulness to and reverence for nature. To be mistaken on this basic point is to make of the martial arts no more than weapons of injury and death.
The very name Aikido indicates its dependence on the laws of nature, which we term ki. Aikido means the way to harmony with ki. That is to say, Aikido is a discipline to make the heart of nature our own heart, to understand love for all things, and to become one with nature. Techniques and physical strength have limits; the great way of the universe stretches to infinity.
Koichi Tohei
Man is infinitely concerned about the infinity to which he belongs, from which he is separated, and for which he is longing. Man is totally concerned about the totality which is his true being and which is disrupted in time and space. Man is unconditionally concerned about that which conditions his being beyond all the conditions in him and around him. Man is ultimately concerned about that which determines his ultimate destiny beyond all preliminary necessities and accidents.
Paul Tillich
Nay, 'tis not fitting that we should require
Within this World but Raiment, Food and Fire;
Powerless Atoms of Eternity
Why should we hope to know of Something higher? This Knowledge could but add, not lessen. Woe;
The Magian who To-day forms fire with snow
Shares with the Sudra in Infinity.
We come from Nothing and to Nothing go. So best consent, although with forced grace,
Upon this dingy Ball to run our race
Untrammeled with the thoughts of higher things,
Until we reach the shadowy Stopping place.
James Branch Cabell
The air is filled with endless images of the objects distributed in it; and all are represented in all, and all in one, and all in each, whence it happens that if two mirrors are placed in such a manner as to face each other exactly, the first will be reflected in the second and the second in the first. The first being reflected in the second takes to it the image of itself with all the images represented in it, among which is the image of the second mirror, and so, image within image, they go on to infinity in such a manner as that each mirror has within it a mirror, each smaller than the last and one inside the other. Thus, by this example, it is clearly proved that every object sends its image to every spot whence the object itself can be seen; and the converse: That the same object may receive in itself all the images of the objects that are in front of it.
Leonardo da Vinci
I have never proceeded from any Genus supremum of the actual infinite. Quite the contrary, I have rigorously proved that there is absolutely no Genus supremum of the actual infinite. What surpasses all that is finite and transfinite is no Genus; it is the single, completely individual unity in which everything is included, which includes the Absolute, incomprehensible to the human understanding. This is the Actus Purissimus, which by many is called God.
I am so in favor of the actual infinite that instead of admitting that Nature abhors it, as is commonly said, I hold that Nature makes frequent use of it everywhere, in order to show more effectively the perfections of its Author. Thus I believe that there is no part of matter which is not - I do not say divisible - but actually divisible; and consequently the least particle ought to be considered as a world full of an infinity of different creatures.
Georg Cantor
No matter how finely you subdivide time and space, each tiny division contains infinity.
But this could imply that you can cut across linear time, open it like a ripe fruit, and see consequential connections. You could be prescient, predict accurately. Predestination and paradox once more.
The flaw must lie in our methods of description, in languages, in social networks of meaning, in moral structures, and in philosophies and religions - all of which convey implicit limits where no limits exist. Paul Muad'Dib, after all, says this time after time throughout Dune.
Frank Herbert
It is not known so generally that Kepler was... a geometrician and algebraist of considerable power, and that he, Desargues, and perhaps Galileo, may be considered as forming a connecting link between the mathematicians of the renaissance and those of modern times. Kepler's work in geometry consists rather in certain general principles enunciated, and illustrated by a few cases, than in any systematic exposition of the subject. In a short chapter on conics inserted in his Paralipomena, published in 1604, he lays down what has been called the principle of continuity, and gives as an example the statement that a parabola is at once the limiting case of an ellipse and of a hyperbola; he illustrates the same doctrine by reference to the foci of conics (the word focus was introduced by him); and he also explains that parallel lines should be regarded as meeting at infinity. He introduced the use of the eccentric angle in discussing properties of the ellipse.
Johannes Kepler
The implications of Descartes' analytic reformulation of geometry are obvious. Not only did the new method make possible a systematic investigation of known curves, but, what is of infinitely greater significance, it potentially created a whole universe of geometric forms beyond conception by the synthetic method.
Descartes also saw that his method applies equally as well to surfaces... but he did not develop this. With the extension to surfaces, there was no reason why geometry should stop with equations in three variables; and the generalization to systems of equations in any finite number of variables was readily made in the nineteenth century. Finally, in the twentieth century, the farthest extension possible in this direction led to spaces of a non-denumerable infinity of dimensions. ...The path from Descartes to the creators of higher space is straight and clear; the remarkable thing is that it was not traveled earlier than it was.
René Descartes
Loneliness is emptiness; and moral nature has as much horror of it as physical nature. Solitude is habitable only for the man of genius who fills it with his ideas - daughters of the spiritual world - or for the beholder of divine works who finds it illuminated by the day of heaven, animated by the breath and by the voice of God. Save these two men, so close to paradise, loneliness is to torture what morale is to physique. Between loneliness and torture there is all the difference between nervous illness and surgical illness. It is suffering multiplied by infinity. The body touches infinity through the nervous system, just as the mind enters it through thought.
Honoré de Balzac
