Diagonal Related Quotes

There was someone called Hippasus in Greek times who found out about the diagonal of a square and they drowned him because no one wanted to know about things like that. Like what? Numbers that make you uncomfortable and don’t relate to oranges.

How can we tell whether the rules which we "guess" at are really right if we cannot analyze the game very well? There are, roughly speaking, three ways. First, there may be situations where nature has arranged, or we arrange nature, to be simple and to have so few parts that we can predict exactly what will happen, and thus we can check how our rules work. (In one corner of the board there may be only a few chess pieces at work, and that we can figure out exactly.) A second good way to check rules is in terms of less specific rules derived from them. For example, the rule on the move of a bishop on a chessboard is that it moves only on the diagonal. One can deduce, no matter how many moves may be made, that a certain bishop will always be on a red square. So, without being able to follow the details, we can always check our idea about the bishop's motion by finding out whether it is always on a red square. Of course it will be, for a long time, until all of a sudden we find that it is on a black square (what happened of course, is that in the meantime it was captured, another pawn crossed for queening, and it turned into a bishop on a black square). That is the way it is in physics. For a long time we will have a rule that works excellently in an over-all way, even when we cannot follow the details, and then some time we may discover a new rule. From the point of view of basic physics, the most interesting phenomena are of course in the new places, the places where the rules do not work—not the places where they do work! That is the way in which we discover new rules. The third way to tell whether our ideas are right is relatively crude but prob-ably the most powerful of them all. That is, by rough approximation. While we may not be able to tell why Alekhine moves this particular piece, perhaps we can roughly understand that he is gathering his pieces around the king to protect it, more or less, since that is the sensible thing to do in the circumstances. In the same way, we can often understand nature, more or less, without being able to see what every little piece is doing, in terms of our understanding of the game.

Another way to picture this is to use Galileo’s ship. Imagine a light beam being shot down from the top of the mast to the deck. To an observer on the ship, the light beam will travel the exact length of the mast. To an observer on land, however, the light beam will travel a diagonal formed by the length of the mast plus the distance (it’s a fast ship) that the ship has traveled forward during the time it took the light to get from the top to the bottom of the mast. To both observers, the speed of light is the same. To the observer on land, it traveled farther before it reached the deck. In other words, the exact same event (a light beam sent from the top of the mast hitting the deck) took longer when viewed by a person on land than by a person on the ship.59 This phenomenon, called time dilation, leads to what is known as the twin paradox. If a man stays on the platform while his twin sister takes off in a spaceship that travels long distances at nearly the speed of light, when she returns she would be younger than he is. But because motion is relative, this seems to present a paradox. The sister on the spaceship might think it’s her brother on earth who is doing the fast traveling, and when they are rejoined she would expect to observe that it was he who did not age much.

The vestibular system tells us about up and down and whether we are upright or not. It tells us where our heads and bodies are in relation to the earth’s surface. It sends sensory messages about balance and movement from the neck, eyes, and body to the CNS for processing and then helps generate muscle tone so we can move smoothly and efficiently. This sense tells us whether we are moving or standing still, and whether objects are moving or motionless in relation to our body. It also informs us what direction we are going in, and how fast we are going. This is extremely useful information should we need to make a fast getaway! Indeed, the fundamental functions of fight, flight, and foraging for food depend on accurate information from the vestibular system. Dr. Ayres writes that the “system has basic survival value at one of the most primitive levels, and such significance is reflected in its role in sensory integration.” The receptors for vestibular sensations are hair cells in the inner ear, which is like a “vestibule” for sensory messages to pass through. The inner-ear receptors work something like a carpenter’s level. They register every movement we make and every change in head position—even the most subtle. Some inner-ear structures receive information about where our head and body are in space when we are motionless, or move slowly, or tilt our head in any linear direction—forward, backward, or to the side. As an example of how this works, stand up in an ordinary biped, or two-footed, position. Now, close your eyes and tip your head way to the right. With your eyes closed, resume your upright posture. Open your eyes. Are you upright again, where you want to be? Your vestibular system did its job. Other structures in the inner ear receive information about the direction and speed of our head and body when we move rapidly in space, on the diagonal or in circles. Stand up and turn around in a circle or two. Do you feel a little dizzy? You should. Your vestibular system tells you instantly when you have had enough of this rotary stimulation. You will probably regain your balance in a moment. What stimulates these inner ear receptors? Gravity! According to Dr. Ayres, gravity is “the most constant and universal force in our lives.” It rules every move we make. Throughout evolution, we have been refining our responses to gravitational pull. Our ancient ancestors, the first fish, developed gravity receptors, on either side of their heads, for three purposes: 1) to keep upright, 2) to provide a sense of their own motions so they could move efficiently, and 3) to detect potentially threatening movements of other creatures through the vibrations of ripples in the water. Millions of years later, we still have gravity receptors to serve the same purposes—except now vibrations come through air rather than water.

Classical number is a thought-process dealing not with spatial relations but with visibly limitable and tangible units, and it follows naturally and necessarily that the Classical knows only the "natural" (positive and whole) numbers, which on the contrary olay in our Western mathematics a quite undistinguished part in the midst of complex, hypercomplex, non-Archimedean and other number-systems. On this account, the idea of irrational numbers the unending decimal fractions of our notation was unrealizable within the Greek spirit. Euclid says and he ought to have been better understood that incommensurable lines are "not related to one another like numbers." In fact, it is the idea of irrational number that, once achieved, separates the notion of number from that of magnitude, for the magnitude of such a number (π, for example) can never be defined or exactly represented by any straight line. Moreover, it follows from this that in considering the relation, say, between diagonal and side in a square the Greek would be brought up suddenly against a quite other sort of number, which was fundamentally alien to the Classical soul, and was consequently feared as a secret of its proper existence too dangerous to be unveiled. There is a singular and significant late-Greek legend, according to which the man who first published the hidden mystery of the irrational perished by shipwreck, "for the unspeakable and the formless must be left hidden for ever".

Classical mathematicians have trouble understanding the set ℝ of Constructive real numbers because it seems to be both countable and uncountable. The Cantor diagonal argument—an algorithm that, given a sequence of real numbers, produces a real number different from every number in the sequence—is completely Constructive, and seems to show that the set is uncountable. But every real number is given by an algorithm that is described by a finite sequence of symbols, and the set of all finite sequences of symbols is countable. The situation clarifies if we see that Cantor discovered a difference in complexity rather than in size. The set ℝ is not bigger, but more complex than the set ℕ. Its complexity is related to the fact that real numbers are algorithms, and to the undecidability of the halting problem shown by Turing. Given a set of symbols purporting to describe the algorithm for a real number, Turing showed that we have no algorithm that decides whether it actually computes a real number or goes into an infinite loop. So we have no way to make a list of all real numbers.

Like Blumenberg, Bataille relates uprightness to the origins of mythology, and, like Freud and Ferenczi, he formats the ‘progressive election [from] quadruped to Homo erectus’ as a deviation from coprophiliac anality. Bataille fixates upon half-upright monkeys, who, he delectates, expose their ‘anal projections’ like ‘excremental skulls’. Inasmuch as their knuckle-dragging existence is some kind of ugly ‘halfway house’ between horizontal and vertical modes of carriage, primates are cast as some kind of partway antithesis on the stepwise ascent to mankind’s upright ‘nobility’: a dialectical step between horizontal and vertical, the monkey is awkwardly diagonal. (Primate posture thus inhabits a kind of uncanny valley—from which Bataille derives much titillation.) Nonetheless, by way of necrotizing the Renaissance cliché of orthograde ‘dignity’, Bataille locates in man’s spinal realignment merely a more refined lasciviousness—a more violent voluptuousness. To wit, he pinpoints ‘Two Terrestrial Axes’: the ‘vertical’, which ‘prolongs the radius of the terrestrial sphere’ as axis of libertine escape, lorded by ocean tides and plants (which ‘flee’ the earth to sacrifice themselves ‘endlessly’ to the Sun’s downward onslaught); and the ‘horizontal’, domicile to beasts and ‘analogous to the turning of the earth’. ‘Only human beings’, Bataille notes, ‘tearing themselves away from peaceful animal horizontality’, have ‘succeeded in appropriating the vegetal erection’, surrendering themselves to exquisite upwards collapse towards outer space’s solar enormities and fluxions.