Mathematics

"... mathematics resembles an iceberg: beneath the surface is the realm of pure mathematics hidden from the public ... Above the water is the tip, the visible part of which we call applied mathematics."

"Proof is the idol before whom the pure mathematician tortures himself."

"Admire the power of the Dirac notation!"

"I do not use Dirac’s bra-ket notation, because for some purposes it is awkward […]"

"Mathematicians are only dealing with the structure of reasoning, and they do not really care what they are talking about. They do not even need to know what they are talking about, or, as they themselves say, whether what they say is true. I will explain that. You state the axioms, such-and-such is so, and such-and-such is so. What then? The logic can be carried out without knowing what the such-and-such words mean. If the statements about the axioms are carefully formulated and complete enough, it is not necessary for the man who is doing the reasoning to have any knowledge of the meaning of the words in order to deduce new conclusions in the same language. … But the physicist has meaning to all his phrases. That is a very important thing that a lot of people who come to physics by way of mathematics do not appreciate. Physics is not mathematics, and mathematics is not physics. One helps the other. But in physics you have to have an understanding of the connection of words with the real world."

"There are at present fundamental problems in theoretical physics awaiting solution, e.g., the relativistic formulation of quantum mechanics and the nature of atomic nuclei (to be followed by more difficult ones such as the problem of life), the solution of which problems will presumably require a more drastic revision of our fundamental concepts than any that have gone before. Quite likely these changes will be so great that it will be beyond the power of human intelligence to get the necessary new ideas by direct attempts to formulate the experimental data in mathematical terms."

"At this point an enigma presents itself which in all ages has agitated inquiring minds. How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality? Is human reason, then, without experience, merely by taking thought, able to fathom the properties of real things? In my opinion the answer to this question is, briefly, this: as far as the propositions of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."

"We cannot deny the indictment that we seek solution for practically every problem of life in quantitative terms, and are not fully aware of the limits of this approach."

"The United States and the post-Stalinist Soviet Union … share the same cultural aims. Both issue from the assumption that wealth is a superior and adequate substitute for symbolic impoverishment. Both American and Soviet cultures are essentially variants of the same belief in wealth as the functional equivalent of a high civilization. In both cultures, the controlling symbolism has been stripped down to belief in the efficacy of wealth. Quantity has become quality. The answer to all questions of “what for?” is “more.”"

"Our culture, obsessed with numbers, has given us the idea that what we can measure is more important than what we can't measure. Think about that for a minute. It means that we make quantity more important than quality."

"It is the same case with all those pretended syllogistical reasonings, which may be found in every other branch of learning, except the sciences of quantity and number; and these may safely, I think, be pronounced the only proper objects of knowledge and demonstration."

"Does it contain any abstract reasoning concerning quantity or number? No. Does it contain any experimental reasoning concerning matter of fact and existence? No. Commit it then to the flames: for it can contain nothing but sophistry and illusion."

"If the quality of society could be replaced by quantity, it would be worth while to live in the world at large; but unfortunately a hundred fools in a crowd still do not produce one intelligent man."

"“Aristocracy,” … taken in its etymological sense, means precisely the power of the elect. The elect, by the very definition of the word, can only be the few, and their power, or rather their authority, being due to their intellectual superiority, has nothing in common with the numerical strength on which democracy is based, a strength whose inherent tendency is to sacrifice the minority to the majority, and therefore quality to quantity and the elect to the masses."

"The mass is a matrix from which all traditional behavior toward works of art issues today in a new form. Quantity has been transmuted into quality. The greatly increased mass of participants has produced a change in the mode of participation."

"Quantity has a quality all its own."

"That curious modern hypostatization “service” is often called in to substitute for the now incomprehensible doctrine of vocation. It tries to secure subordination by hypothesizing something larger than the self, which turns out, however, to be only a multitude of selfish selves. The familiar change from quality to quantity may again be noted; one serves not the higher part of the self (this entails hierarchy) … but merely consumer demand. And who admires those at the top of a hierarchy of consumption? Man as a consuming animal is thus seen to be not enough."

"The demon of quantity, who will soon rule the world, is pressing home his attack and fortifying his positions in literary circles as everywhere else."

"Monotony of evil: never anything new, everything about it is equivalent. ... It is because of this monotony that quantity plays so great a part. A host of women (Don Juan) or of men (Célimène), etc."

"The spirit, overcome by the weight of quantity, has no longer any other criterion than efficiency."

"Motion in three dimensions is not easy to understand. For example, you are probably good at driving a car along a freeway (one-dimensional motion) but would probably have a difficult time in landing an airplane on a runway (three dimensional motion) without a lot of training."

"Perhaps the first to approach the fourth dimension from the side of physics, was the Frenchman, Nicole Oresme, of the fourteenth century. In a manuscript treatise, he sought a graphic representation of the Aristotelian forms, such as heat, velocity, sweetness, by laying down a line as a basis designated longitudo, and taking one of the forms to be represented by lines (straight or circular) perpendicular to this either as a latitudo or an altitudo. The form was thus represented graphically by a surface. Oresme extended this process by taking a surface as the basis which, together with the latitudo, formed a solid. Proceeding still further, he took a solid as a basis and upon each point of this solid he entered the increment. He saw that this process demanded a fourth dimension which he rejected; he overcame the difficulty by dividing the solid into numberless planes and treating each plane in the same manner as the plane above, thereby obtaining an infinite number of solids which reached over each other. He uses the phrase "fourth dimension" (4am dimensionem)."

"I search for the realness, the real feeling of a subject, all the texture around it... I always want to see the third dimension of something... I want to come alive with the object."

"There is no difference between Time and any of the three dimensions of Space except that our consciousness moves along it."

"When you listen to a thought, you are aware not only of the thought but also of yourself as the witness of the thought. A new dimension of consciousness has come in."

"Make it your practice to withdraw attention from past and future whenever they are not needed. Step out of the time dimension as much as possible in everyday life."

"He sat back down again in his armchair. “Of course, that name really isn’t accurate. I suppose a pentaract should really be a four-dimensional pentagon, and this is meant to be a picture of a five-dimensional cube.”"

"You unlock this door with the key of imagination. Beyond it is another dimension—a dimension of sound, a dimension of sight, a dimension of mind. You're moving into a land of both shadow and substance, of things and ideas. You've just crossed over into the Twilight Zone."

"In a mathematical sense, space is manifoldness, or combination of numbers. Physical space is known as the 3-dimension system. There is the 4-dimension system, there is the 10-dimension system."

"Well, I do not mind telling you I have been at work upon this geometry of Four Dimensions for some time. Some of my results are curious. For instance, here is a portrait of a man at eight years old, another at fifteen, another at seventeen, another at twenty-three, and so on. All these are evidently sections, as it were, Three-Dimensional representations of his Four-Dimensioned being, which is a fixed and unalterable thing."

"You are traveling through another dimension, a dimension not only of sight and sound but of mind. A journey into a wondrous land whose boundaries are that of imagination. Your next stop, the Twilight Zone!"

"You are about to enter another dimension, a dimension not only of sight and sound but of mind. A journey into a wondrous land of imagination. Next stop, the Twilight Zone!"

"You're traveling through another dimension, a dimension not only of sight and sound but of mind. A journey into a wondrous land of imagination. Next stop, the Twilight zone!"

"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."

"Are you trying to tell me that this gadget's got a fourth dimensional extension?" Paradine demanded. "Not visually, anyway," Holloway denied. "All I say is that our minds, conditioned to Euclid, can see nothing in this but an illogical tangle of wires. But a child especially a baby might see more. Not at first. It'd be a puzzle, of course. Only a child wouldn't be handicapped by too many preconceived ideas."

"If I am recalling an incident very vividly I go back to the instant of its occurrence; I become absent minded, as you say. I jump back for a moment. Of course we have no means of staying back for any length of time any more than a savage or an animal has of staying six feet above the ground. But a civilized man is better off than the savage in this respect. He can go up against gravitation in a balloon, and why should we not hope that ultimately he may be able to stop or accelerate his drift along the Time Dimension; or even to turn about and travel the other way?"

"We are not in the Eighth dimension, we are over New Jersey. Hope is not lost."

"You're traveling through another dimension, a dimension not only of sight and sound but of mind. A journey into a wondrous land whose boundaries are that of imagination. That's the signpost up ahead — your next stop, the Twilight Zone!"

"Time is relative, it can stretch and squeeze, but it can't run backwards. The only thing that can move across dimensions like time is gravity. … Look, Cooper, they're creatures of at least five dimensions, to them the past might be a canyon they can climb into and the future a mountain they can climb up... but to us it's not, okay?"

"How sickness enlarges the dimension of a man's self to himself!"

"Design in art, is a recognition of the relation between various things, various elements in the creative flux. You can't invent a design. You recognize it, in the fourth dimension. That is, with your blood and your bones, as well as with your eyes."

"We actually have a candidate for the mind of God. The mind of God we believe is cosmic music, the music of strings resonating through 11 dimensional hyperspace. That is the mind of God."

"The supremacy of expediency is being refuted by time and truth. Time is an essential dimension of existence defiant of man's power, and truth reigns in supreme majesty, unrivaled, inimitable, and can never be defeated."

"Faith is sensitiveness to what transcends nature, knowledge and will, awareness of the ultimate, alertness to the holy dimension of all reality. Faith is a force in man, lying deeper than the stratum of reason and its nature cannot be defined in abstract, static terms. To have faith is not to infer the beyond from the wretched here, but to perceive the wonder that is here and to be stirred by the desire to integrate the self into the holy order of living. It is not a deduction but an intuition, not a form of knowledge, of being convinced without proof, but the attitude of mind toward ideas whose scope is wider than its own capacity to grasp. Such alertness grows from the sense for the meaningful, for the marvel of matter, for the core of thoughts. It is begotten in passionate love for the significance of all reality, in devotion to the ultimate meaning which is only God."

"Science by itself has no moral dimension. But it does seek to establish truth. And upon this truth morality can be built."

"Human rights, of course, must include the right to religious freedom, understood as the expression of a dimension that is at once individual and communitarian — a vision that brings out the unity of the person while clearly distinguishing between the dimension of the citizen and that of the believer."

"Space has always been the spiritual dimension of architecture. It is not the physical statement of the structure so much as what it contains that moves us."

"Imaginary time is a new dimension, at right angles to ordinary, real time."

"In a sensible theory there are no [dimensionless] numbers whose values are determinable only empirically. I can, of course, not prove that ... dimensionless constants in the laws of nature, which from a purely logical point of view can just as well have other values, should not exist. To me in my 'Gottvertrauen' [faith in God] this seems evident, but there might be few who have the same opinion ..."

"We have created characters and animated them in the dimension of depth, revealing through them to our perturbed world that the things we have in common far outnumber and outweigh those that divide us."