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| Link | Quote | Stars | Tags | Author |
| db1512a | His ranch adjoined the Rio Grande, the river that formed the boundary between the United States and Mexico, with its poverty, caste system, and systemic corruption. So the poor Mexicans migrated. Over thirteen million of them, over a fifth of the Mexican population, had crossed that border illegally in the last fifty years and were grubbing for work in the United States, usually for minimum wage, or living on welfare and food stamps. Illite.. | Stephen Coonts | ||
| 9a06262 | American politicians had done little through the years to stem the flood. Hispanic voters wanted their kinsmen to be able to enter the United States regardless of their ability to contribute to the economy or pay their own bills, yet this wasn't the decisive factor. Farmers and small-business men wanted a source of cheap labor, and were content to pass the true costs, the social costs, on to the taxpayers. Generous public welfare programs a.. | Stephen Coonts | ||
| 6fe1571 | God has fixed the time for my death. I do not concern myself about that, but to be always ready, no matter when it may overtake me. That is the way all men should live, and then all would be equally brave. | Stephen Coonts | ||
| fe13cee | Medicine often tastes bad, but until we fix the government policies that breed poverty, we have condemned the poor, black, white, and brown to a life of economic slavery. | Stephen Coonts | ||
| 81f739d | If there were no love there would be no hatred. Just sorrow. And | Stephen Coonts | ||
| f4cc638 | I've talked to Molina three times today, and he wants me to pull the rabbit out of the hat now"." | Stephen Coonts | ||
| ae46407 | When one has a text to question, it is irrelevant to ask the author. | Umberto Eco | ||
| aa6ca55 | Communication is pointless and we're all doomed. | Frank Portman | ||
| d3e81f7 | I'm not any religion myself, but for the record, I'm pretty sure I do believe in God. It's just a feeling I have. I can't prove it, but since when are you supposed to prove a feeling? God is the only situation where they expect you to do that. Plus, God embarrasses people. Which I totally enjoy. | Frank Portman | ||
| 0645b6f | Dance Dance Revolution was, however, no enticement whatsoever to one who had tasted the delights of the King of Sacramento's shadowed chamber. | Frank Portman | ||
| d8d98b4 | We may be the only people in the world who can say our goal is to have people leave our website as quickly as possible. --"Ten Things We Know to Be True," Google company website" | George Beahm | ||
| de1ba9b | My job is not to be easy on people. My job is to make them better. | Beahm George | ||
| 8f9884f | Specifically, the awareness that I claim is demonstrably non-computational is our understanding of the properties of natural numbers 0,1,2,3,4,....(One might even say that our concept of a natural number is, in a sense, a form of non-geometric 'visualization'.) We shall see in 2.5, by a readily accessible form of Godel's theorem (cf. response to query Q16), that this understanding is something that cannot be simulated computationally. From .. | Roger Penrose | ||
| c99aa8c | It is a common misconception, in the spirit of the sentiments expressed in Q16, that Godel's theorem shows that there are many different kinds of arithmetic, each of which is equally valid. The particular arithmetic that we may happen to choose to work with would, accordingly, be defined merely by some arbitrarily chosen formal system. Godel's theorem shows that none of these formal systems, if consistent, can be complete; so-it is argued-w.. | Roger Penrose | ||
| 344e400 | Jacques was so impressed with the beauty of the curve known as a logarithmic spiral (Figure 37; the name was derived from the way in which the radius grows as we move around the curve clockwise) that he asked that this shape, and the motto he assigned to it: "Eadem mutato resurgo" (although changed, I rise again the same), be engraved on his tombstone. The motto describes a fundamental property unique to the logarithmic spiral-it does not a.. | Mario Livio | ||
| 0d8b90d | Q5. Have not I merely shown that it is possible to outdo just a particular algorithmic procedure, A, by defeating it with the computation Cq(n)? Why does this show that I can do better than any A whatsoever? The argument certainly does show that we can do better than any algorithm. This is the whole point of a reductio ad absurdum argument of this kind that I have used here. I think that an analogy might be helpful here. Some readers will k.. | Roger Penrose | ||
| 5716b5f | The reason that I have concentrated on non-computability, in my arguments, rather than on complexity, is simply that it is only with the former that I have been able to see how to make the necessary strong statements. It may well be that in the working lives of most mathematicians, non-computability issues play, if anything, only a very small part. But that is not the point at issue. I am trying to show that (mathematical) understanding is .. | Roger Penrose | ||
| 0c36bfe | In order for A to apply to computations generally, we shall need a way of coding all the different computations C(n) so that A can use this coding for its action. All the possible different computations C can in fact be listed, say as C0, C1, C2, C3, C4, C5,..., and we can refer to Cq as the qth computation. When such a computation is applied to a particular number n, we shall write C0(n), C1(n), C2(n), C3(n), C4(n), C5(n),.... We can take .. | Roger Penrose | ||
| ab28dc8 | What Godel and Rosser showed is that the consistency of a (sufficiently extensive) formal system is something that lies outside the power of the formal system itself to establish. | Roger Penrose | ||
| b3420ea | The thrust of Godel's argument for our purposes is that it shows us how to go beyond any given set of computational rules that we believe to be sound, and obtain a further rule, not contained in those rules, that we must believe to be sound also, namely the rule asserting the consistency of the original rules. The essential point, for our purposes, is: belief in soundness implies belief in consistency. We have no right to use the rules of a.. | Roger Penrose | ||
| 10a5a3f | If, as I believe, the Godel argument is consequently forcing us into an acceptance of some form of viewpoint C, the we shall also have to come to terms with some of its other implications. We shall find ourselves driven towards a Platonic viewpoint of things. According to Plato, mathematical concepts and mathematical truths inhabit an actual world of their own that is timeless and without physical location. Plato's world is an ideal world o.. | Roger Penrose | ||
| f28430c | Mathematical truth is not determined arbitrarily by the rules of some 'man-made' formal system, but has an absolute nature, and lies beyond any such system of specifiable rules. Support for the Platonic viewpoint ...was an important part of Godel's initial motivations. | Roger Penrose | ||
| ff5b846 | Supporters of the "modified Platonic view" of mathematics like to point out that, over the centuries, mathematicians have produced (or "discovered") numerous objects of pure mathematics with absolutely no application in mind. Decades later, these mathematical constructs and models were found to provide solutions to problems in physics. Penrose tilings and non-Euclidean geometries are beautiful testimonies to this process of mathematics unex.. | Mario Livio | ||
| 7e20552 | The curriculum for the education of statesmen at the time of Plato included arithmetic, geometry, solid geometry, astronomy, and music-all of which, the Pythagorean Archytas tells us, fell under the general definition of "mathematics." According to legend, when Alexander the Great asked his teacher Menaechmus (who is reputed to have discovered the curves of the ellipse, the parabola, and the hyperbola) for a shortcut to geometry, he got the.. | Mario Livio | ||
| 1d21842 | Nature loves logarithmic spirals. From sunflowers, seashells, and whirlpools, to hurricanes and giant spiral galaxies, it seems that nature chose this marvelous shape as its favorite "ornament." The constant shape of the logarithmic spiral on all size scales reveals itself beautifully in nature in the shapes of minuscule fossils or unicellular organisms known as foraminifera. Although the spiral shells in this care are composite structures .. | Mario Livio | ||
| 947cb28 | Gell-Mann and Ne'eman discovered that one such simple Lie group, called "special unitary group of degree 3," or SU(3), was particularly well suited for the "eightfold way"-the family structure the particles were found to obey. The beaty of the SU(3) symmetry was revealed in full glory via its predictive power. Gell-Mann and Ne'eman showed that if the theory were to hold true, a previously unknown tenth member of a particular family of nine .. | Mario Livio | ||
| ec8f06f | As we shall see throughout this book, the unifying powers of group theory are so colossal that historian of mathematics Eric Temple Bell (1883-1960) once commented, "When ever groups disclosed themselves, or could be introduced, simplicity crystallized out of comparative chaos." | Mario Livio | ||
| 15a240f | In the late 1960's, physicists Steven Weinberg, Abdus Salam, and Sheldon Glashow conquered the next unification frontier. In a phenomenal piece of scientific work they showed that the electromagnetic and weak nuclear forces are nothing but different aspects of the same force, subsequently dubbed the electroweak force. The predictions of the new theory were dramatic. The electromagnetic force is produced when electrically charged particles e.. | Mario Livio | ||
| 026d1a7 | In an old joke, a physicist and a mathematician are asked what they would do if they needed to iron their pants, but although they are in possession of an iron, the electric outlet is in the adjacent room. Both answer that they would take the iron to the second room and plug it in there. Now they are asked what they would do if they were already in the room in which the outlet is located. They physicist answers that he would plug the iron i.. | Mario Livio | ||
| 41080c2 | There were also many cases of feedback between physics and mathematics, where a physical phenomenon inspired a mathematical model that later proved to be the explanation of an entirely different physical phenomenon. An excellent example is provided by the phenomenon known as Brownian motion. In 1827, British botanist Robert Brown (1773-1858) observed that wen pollen particles are suspended in water, they get into a state of agitated motion... | Mario Livio | ||
| a84f978 | Is it odd how asymmetrical Is "symmetry"? "Symmetry" is asymmetrical. How odd it is. This stanza remains unchanged if read word by word from the end to the beginning-it is symmetrical with respect to backward reading." | Mario Livio | ||
| 69049e5 | Through the works of Weinberg, Glashow, and Salam on the electroweak theory and the elegant framework developed by the physicists David Gross, David Politzer, and Frank Wilczek for quantum chromodynamics, the characteristic group of the standard model has been identified with a product of three Lie groups denoted by U(1), SU(2), and SU(3). In some sense, therefore, the road toward the ultimate unification of the forces of nature has to go t.. | Mario Livio | ||
| 41668e3 | Even though it is almost impossible to attribute with certainty any specific mathematical achievements either to Pythagoras himself or to his followers, there is no question that they have been responsible for a mingling of mathematics, philosophy of life, and religion unparalleled in history. In this respect it is perhaps interesting to note the historical coincidence that Pythagoras was a contemporary of Buddha and Confucius. | Mario Livio | ||
| 593fc43 | We have already seen that gauge symmetry that characterizes the electroweak force-the freedom to interchange electrons and neturinos-dictates the existence of the messenger electroweak fields (photon, W, and Z). Similarly, the gauge color symmetry requires the presence of eight gluon fields. The gluons are the messengers of the strong force that binds quarks together to form composite particles such as the proton. Incidentally, the color "c.. | Mario Livio | ||
| 6323c69 | The properties that define a group are: 1. Closure. The offspring of any two members combined by the operation must itself be a member. In the group of integers, the sum of any two integers is also an integer (e.g., 3 + 5 = 8). 2. Associativity. The operation must be associative-when combining (by the operation) three ordered members, you may combine any two of them first, and the result is the same, unaffected by the way they are bracketed.. | Mario Livio | ||
| 680e445 | The importance of mirror-reflection symmetry to our perception and aesthetic appreciation, to the mathematical theory of symmetries, to the laws of physics, and to science in general, cannot be overemphasized, and I will return to it several times. Other symmetries do exist, however, and they are equally relevant. | Mario Livio | ||
| b003c15 | In other words, Birkhoff proposed a formula for the feeling of aesthetic value: M = O / C. The meaning of this formula is: For a given degree of complexity, the aesthetic measure is higher the more order the object possesses. Alternatively, if the amount of order is specified, the aesthetic measure is higher the less complex the object. Since for most practical purposes, the order is determined primarily by the symmetries of the object, Bir.. | Mario Livio | ||
| b6b676f | An interesting question is whether symmetry with respect to translation, and indeed reflection and rotation too, is limited to the visual arts, or may be exhibited by other artistic forms, such as pieces of music. Evidently, if we refer to the sounds, rather than to the layout of the written musical score, we would have to define symmetry operations in terms other than purely geometrical, just as we did in the case of the palindromes. Once .. | Mario Livio | ||
| 7101a4e | Our mathematics is a combination of invention and discoveries. The axioms of Euclidean geometry as a concept were an invention, just as the rules of chess were an invention. The axioms were also supplemented by a variety of invented concepts, such as triangles, parallelograms, ellipses, the golden ratio, and so on. The theorems of Euclidean geometry, on the other hand, were by and large discoveries; they were the paths linking the different.. | Mario Livio | ||
| 874f7f8 | The erosion of trust in public school systems has had catastrophic consequences, and will take decades to put right. As we've seen, attempts to make schools 'more accountable' for their test scores leave teachers torn between what psychologist Barry Schwartz calls 'doing the right thing and doing the required thing'. The right thing is to teach students through personalised, flexible methods, according to their needs, interests and aspirati.. | David Price | ||
| 31a55be | So the researchers concluded that being forced to confront trade-offs in making decisions makes people unhappy and indecisive. | Barry Schwartz | ||
| da51e33 | Adding the second option creates a conflict, forcing a trade-off between price and quality. | Barry Schwartz | ||
| 517fbcf | So it seems that neither our predictions about how we will feel after an experience nor our memories of how we did feel during the experience are very accurate reflections of how we actually do feel while the experience is occurring. And yet it is memories of the past and expectations for the future that govern our choices. | Barry Schwartz | ||
| 5f7cc9a | emotional cost of potential trade-offs does more than just diminish our sense of satisfaction with a decision. It also interferes with the quality of decisions themselves. | Barry Schwartz |