Algebraic Related Quotes

Algebra applies to the clouds, the radiance of the star benefits the rose--no thinker would dare to say that the perfume of the hawthorn is useless to the constellations. Who could ever calculate the path of a molecule? How do we know that the creations of worlds are not determined by falling grains of sand? Who can understand the reciprocal ebb and flow of the infinitely great and the infinitely small, the echoing of causes in the abyss of being and the avalanches of creation? A mite has value; the small is great, the great is small. All is balanced in necessity; frightening vision for the mind. There are marvelous relations between beings and things, in this inexhaustible whole, from sun to grub, there is no scorn, each needs the other. Light does not carry terrestrial perfumes into the azure depths without knowing what it does with them; night distributes the stellar essence to the sleeping plants. Every bird that flies has the thread of the infinite in its claw. Germination includes the hatching of a meteor and the tap of a swallow's beak breaking the egg, and it guides the birth of the earthworm, and the advent of Socrates. Where the telescope ends, the microscope begins. Which of the two has a greater view? Choose. A bit of mold is a pleiad of flowers; a nebula is an anthill of stars. The same promiscuity, and still more wonderful, between the things of the intellect and material things. Elements and principles are mingled, combined, espoused, multiplied one by another, to the point that the material world, and the moral world are brought into the same light. Phenomena are perpetually folded back on themselves. In the vast cosmic changes, universal life comes and goes in unknown quantities, rolling everything up in the invisible mystery of the emanations, using everything, losing no dream from any single sleep, sowing a microscopic animal here, crumbling a star there, oscillating and gyrating, making a force of light, and an element of thought, disseminated and indivisible dissolving all, that geometric point, the self; reducing everything to the soul-atom; making everything blossom into God; entangling from the highest to the lowest, all activities in the obscurity of a dizzying mechanism, linking the flight of an insect to the movement of the earth, subordinating--who knows, if only by the identity of the law--the evolutions of the comet in the firmament to the circling of the protozoa in the drop of water. A machine made of mind. Enormous gearing, whose first motor is the gnat, and whose last is the zodiac.

Algebra was far more interesting when it was a matter of proportioning out mutton chops so as to poison only half of one’s dinner guests and then determining the relative value of purchasing a more expensive, yet more effective, antidote over a home remedy.

Each week I plot your equations dot for dot, xs against ys in all manner of algebraical relation, and every week they draw themselves as commonplace geometry, as if the world of forms were nothing but arcs and angles. God’s truth, Septimus, if there is an equation for a curve like a bell, there must be an equation for one like a bluebell, and if a bluebell, why not a rose? Do we believe nature is written in numbers? Septimus We do. Thomasina Then why do your equations only describe the shapes of manufacture? Septimus I do not know. Thomasina Armed thus, God could only make a cabinet.

Visual thinkers, on the other hand, see images in their mind’s eye that allow them to make rapid-fire associations. Generally, visual thinkers like maps, art, and mazes, and often don’t need directions at all. Some visual thinkers can easily locate a place they’ve been to only once, their internal GPS having logged the visual landmarks. Visual thinkers tend to be late talkers who struggle with school and traditional teaching methods. Algebra is often their undoing, because the concepts are too abstract, with little or nothing concrete to visualize. Visual thinkers tend to be good at arithmetic that is directly related to practical

We also find *physics*, in the widest sense of the word, concerned with the explanation of phenomena in the world; but it lies already in the nature of the explanations themselves that they cannot be sufficient. *Physics* is unable to stand on its own feet, but needs a *metaphysics* on which to support itself, whatever fine airs it may assume towards the latter. For it explains phenomena by something still more unknown than are they, namely by laws of nature resting on forces of nature, one of which is also the vital force. Certainly the whole present condition of all things in the world or in nature must necessarily be capable of explanation from purely physical causes. But such an explanation―supposing one actually succeeded so far as to be able to give it―must always just as necessarily be burdened with two essential imperfections (as it were with two sore points, or like Achilles with the vulnerable heel, or the devil with the cloven foot). On account of these imperfections, everything so explained would still really remain unexplained. The first imperfection is that the *beginning* of the chain of causes and effects that explains everything, in other words, of the connected and continuous changes, can positively *never* be reached, but, just like the limits of the world in space and time, recedes incessantly and *in infinitum*. The second imperfection is that all the efficient causes from which everything is explained always rest on something wholly inexplicable, that is, on the original *qualities* of things and the *natural forces* that make their appearance in them. By virtue of such forces they produce a definite effect, e.g., weight, hardness, impact, elasticity, heat, electricity, chemical forces, and so on, and such forces remain in every given explanation like an unknown quantity, not to be eliminated at all, in an otherwise perfectly solved algebraical equation. Accordingly there is not a fragment of clay, however little its value, that is not entirely composed of inexplicable qualities. Therefore these two inevitable defects in every purely physical, i.e., causal, explanation indicate that such an explanation can be only *relatively* true, and that its whole method and nature cannot be the only, the ultimate and hence sufficient one, in other words, cannot be the method that will ever be able to lead to the satisfactory solution of the difficult riddles of things, and to the true understanding of the world and of existence; but that the *physical* explanation, in general and as such, still requires one that is *metaphysical*, which would furnish the key to all its assumptions, but for that very reason would have to follow quite a different path. The first step to this is that we should bring to distinct consciousness and firmly retain the distinction between the two, that is, the difference between *physics* and *metaphysics*. In general this difference rests on the Kantian distinction between *phenomenon* and *thing-in-itself*. Just because Kant declared the thing-in-itself to be absolutely unknowable, there was, according to him, no *metaphysics* at all, but merely immanent knowledge, in other words mere *physics*, which can always speak only of phenomena, and together with this a critique of reason which aspires to metaphysics." ―from_The World as Will and Representation_. Translated from the German by E. F. J. Payne. In Two Volumes, Volume II, pp. 172-173

Hume begins by distinguishing seven kinds of philosophical relation: resemblance, identity, relations of time and place, proportion in quantity or number, degrees in any quality, contrariety, and causation. These, he says, may be divided into two kinds: those that depend only on the ideas, and those that can be changed without any change in the ideas. Of the first kind are resemblance, contrariety, degrees in quality, and proportions in quantity or number. But spatio-temporal and causal relations are of the second kind. Only relations of the first kind give certain knowledge; our knowledge concerning the others is only probable. Algebra and arithmetic are the only sciences in which we can carry on a long chain of reasoning without losing certainty. Geometry is not so certain as algebra and arithmetic, because we cannot be sure of the truth of its axioms. It is a mistake to suppose, as many philosophers do, that the ideas of mathematics 'must be comprehended by a pure and intellectual view, of which the superior faculties of the soul are alone capable'. The falsehood of this view is evident, says Hume, as soon as we remember that 'all our ideas are copied from our impressions'. The three relations that depend not only on ideas are identity, spatio-temporal relations, and causation. In the first two, the mind does not go beyond what is immediately present to the senses. (Spatio-temporal relations, Hume holds, can be perceived, and can form parts of impressions.) Causation alone enables us to infer some thing or occurrence from some other thing or occurrence: "'Tis only causation, which produces such a connexion, as to give us assurance from the existence or action of one object, that 'twas followed or preceded by any other existence or action.

We can understand why one of the titles given to Jesus is that of ‘prophet.’ Jesus is the last and greatest of the prophets, the one who sums them up and goes further than all of them. He is the prophet of the last, but also of the best, chance. With him there takes place a shift that is both tiny and gigantic – a shift that follows on directly from the Old Testament but constitutes a decisive break as well. This is the complete elimination of the sacrificial for the first time – the end of divine violence and the explicit revelation of all that has gone before. It calls for a complete change of emphasis and a spiritual metamorphosis without precedent in the whole history of mankind. It also amounts to an absolute simplification of the relations between human beings, in so far as all the false differences between doubles are annulled – a simplification in the sense in which we speak of an algebraic simplification. Throughout the texts of the Old Testament it was impossible to conclude the deconstruction of myths, rituals and law since the plenary revelation of the founding murder had not yet taken place. The divinity may be to some extent stripped of violence, but not completely so. That is why there is still an indeterminate and indistinct future, in which the resolution of the problem by human means alone – the face-to-face reconciliation that ought to result when people are alerted to the stupidity and uselessness of symmetrical violence – remains confused to a certain extent with the hope of a new epiphany of violence that is distinctively divine in origin, a ‘Day of Yahweh’ that would combine the paroxysm of God’s anger with a no less God-given reconciliation. However remarkably the prophets progress toward a precise understanding of what it is that structures religion and culture, the Old Testament never tips over into the complete rationality that would dispense with this hope of a purgation by violence and would give up requiring God to take the apocalyptic solution by completely liquidating the ‘evil’ in order to ensure the happiness of the chosen.

To wit, researchers recruited a large group of college students for a seven-day study. The participants were assigned to one of three experimental conditions. On day 1, all the participants learned a novel, artificial grammar, rather like learning a new computer coding language or a new form of algebra. It was just the type of memory task that REM sleep is known to promote. Everyone learned the new material to a high degree of proficiency on that first day—around 90 percent accuracy. Then, a week later, the participants were tested to see how much of that information had been solidified by the six nights of intervening sleep. What distinguished the three groups was the type of sleep they had. In the first group—the control condition—participants were allowed to sleep naturally and fully for all intervening nights. In the second group, the experimenters got the students a little drunk just before bed on the first night after daytime learning. They loaded up the participants with two to three shots of vodka mixed with orange juice, standardizing the specific blood alcohol amount on the basis of gender and body weight. In the third group, they allowed the participants to sleep naturally on the first and even the second night after learning, and then got them similarly drunk before bed on night 3. Note that all three groups learned the material on day 1 while sober, and were tested while sober on day 7. This way, any difference in memory among the three groups could not be explained by the direct effects of alcohol on memory formation or later recall, but must be due to the disruption of the memory facilitation that occurred in between. On day 7, participants in the control condition remembered everything they had originally learned, even showing an enhancement of abstraction and retention of knowledge relative to initial levels of learning, just as we’d expect from good sleep. In contrast, those who had their sleep laced with alcohol on the first night after learning suffered what can conservatively be described as partial amnesia seven days later, forgetting more than 50 percent of all that original knowledge. This fits well with evidence we discussed earlier: that of the brain’s non-negotiable requirement for sleep the first night after learning for the purposes of memory processing. The real surprise came in the results of the third group of participants. Despite getting two full nights of natural sleep after initial learning, having their sleep doused with alcohol on the third night still resulted in almost the same degree of amnesia—40 percent of the knowledge they had worked so hard to establish on day 1 was forgotten.

the simple algebraic equation ω+k3 = 0. This is called the dispersion relation of (1): with the help of the Fourier transform it is not hard to show that every solution is a superposition of solutions of the form ei(kx-ωt), and the dispersion relation tells us how the “wave number” k is related to the “angular frequency” ω in each of these elementary solutions.

During their first lesson, as Radu had feverishly scrambled to keep up and Mehmed had recited whole sections of the Koran, Lada spoke only in Wallachian. Molla Gurani had merely gazed at her, impassive behind those hated lenses, and informed her that his sole duty was to educate Mehmed. And, he had added in a disinterested tone, I do not think women capable of much learning. It is to do with the shape of their heads. Lada excelled after that. She memorized more sections of the Koran than either of the boys, and intoned them in a mocking imitation of Molla Gurani. She completed every theorem and practice of mathematic and algebraic problems. She knew the history of the Ottoman state and Mehmed’s line of descent as well as Mehmed himself. Mehmed was nearly thirteen, born between Lada and Radu. He was a third son, his mother a slave concubine, and his father favored the eldest two sons, which subjected Mehmed to gossip and shame. It was dreary knowledge, and Lada worked hard not to relate to or pity Mehmed. But above all, more than any other subject, she devoured lessons on past battles, historical alliances, and border disputes. For a while she had feared that Molla Gurani had meant to trick her into studiousness with his challenge, but he remained as impassive as ever, showing no pleasure in her attentiveness, never rising to her baiting. It did, however, greatly chagrin Mehmed whenever she surpassed him. That became her new goal.

In his first class of the day, correlated language arts, a class for students at least two years below their grade level in English, Boobie Miles spent the period working on a short research paper that he called “The Wonderful Life of Zebras.” He thumbed through various basic encyclopedia entries on the zebra. He ogled at how fast they ran (“Damn, they travel thirty miles”) and was so captivated by a picture of a zebra giving birth that he showed it to a classmate (“Want to see it have a baby, man?”). By the end of the class, Boobie produced the following thesis paragraph: Zebras are one of the most unusual animals in the world today. The zebra has many different kind in it nature. The habitat of the zebra is in wide open plain. Many zebras have viris types of relatives. He then went on to algebra I, a course that the average college-bound student took in ninth grade and some took in eighth. Because of his status as a special needs student, Boobie hadn’t taken the course until his senior year. He was having difficulty with it and his average midway through the fall was 71. After lunch it was on to creative writing, where Boobie spent a few minutes playing with a purple plastic gargoyle-looking monster. He lifted the fingers of the monster so it could pick its nose, then stuck his own fingers into its mouth. There were five minutes of instruction that day; students spent the remaining fifty-odd minutes working on various stories they were writing. They pretty much could do what they wanted. Boobie wrote a little and also explained to two blond-haired girls what some rap terms meant, that “chillin’ to the strength,” for example, meant “like cool to the max.” Boobie enjoyed this class. It gave him an unfettered opportunity to express himself, and the teacher didn’t expect much from him. His whole purpose in life, she felt, was to be a football player. “That’s the only thing kids like that have going for them, is that physical strength,” she said.

Note, however, that as conventionally understood, the operational and relational closure of an algebraic system does not imply that its identity is completely self-contained. Instead, the mathematician and any required display, storage, or processing media are typically missing from its formal idealization, and it is usually considered to exist in a Platonic realm beyond which no explanation is required. In a ToE, this is unacceptable.

Predication may be unconventionally, but not really inaccurately, defined as, 'Whatever is done by the word is in such a sentence as: a horse is an animal; the earth is a planet.' If I say a horse is an animal; then a) if by the word animal I mean something more, or less, or other than horse, I have told a lie; but b) if I do not mean by the word animal something more, or less, or other than horse, I have said almost nothing. For I might was well have said a horse is a horse. Hence the attempts we are now witnessing to replace the traditional logic based on predication by a new logic, in which symbols of algebraic precision refer to 'atomic' facts and events having no vestige of connection with the symbols and no hierarchical relation to each other.

Descartes arrives at four precepts that “would prove perfectly sufficient for me, provided I took the firm and unwavering resolution never in a single instance to fail in observing them.” They amount to a kind of diagram for how to think. He writes: The first was never to accept anything for true which I did not clearly know to be such … to comprise nothing more in my judgment than what was presented to my mind so clearly and distinctly as to exclude all ground of doubt. The second, to divide each of the difficulties under examination into as many parts as possible, and as might be necessary for its adequate solution. The third, to conduct my thoughts in such order that, by commencing with objects the simplest and easiest to know, I might ascend by little and little, and, as it were, step by step, to the knowledge of the more complex; assigning in thought a certain order even to those objects which in their own nature do not stand in a relation of antecedence and sequence. And the last, in every case to make enumerations so complete, and reviews so general, that I might be assured that nothing was omitted.

Living on the Plains” That winter when this thought came-how the river held still every midnight and flowed backward a minute-we studied algebra late in our room fixed up in the barn, and I would feel the curved relation, the rafters upside down, and the cows in their life holding the earth round and ready to meet itself again when morning came. At breakfast while my mother stirred the cereal she said, "You're studying too hard," and I would include her face and hands in my glance and then look past my father's gaze as he told again our great race through the stars and how the world can't keep up with our dreams.