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Either because of mate selection, cognition, predator avoidance, or a combination of all three, our minds are attracted to and are finely tuned to the detection of symmetry. The question of whether symmetry is truly fundamental to the universe itself, or merely to the universe as perceived by humans, thus becomes particularly acute.
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Mario Livio |
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a60f988
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Computer scientist and author Douglas R. Hofstadter phrased this succinctly in his fantastic book Godel, Escher, Bach: An Eternal Golden Braid: "Provability is a weaker notion than truth." In this sense, there will never be a formal method of determining for every mathematical proposition whether it is absolutely true, any more than there is a way to determine whether a theory in physics is absolutely true. Oxford's mathematical physicist R..
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Mario Livio |
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9920395
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The legendary inscription above the Academy's door speaks loudly about Plato's attitude toward mathematics. In fact, most of the significant mathematical research of the fourth century BC was carried out by people associated in one way or another with the Academy. Yet Plato himself was not a mathematician of great technical dexterity, and his direct contributions to mathematical knowledge were probably minimal. Rather, he was an enthusiasti..
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Mario Livio |
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6f4914b
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The Greek excellence in mathematics was largely a direct consequence of their passion for knowledge for its own sake, rather than merely for practical purposes. A story has it that when a student who learned one geometrical proposition with Euclid asked, "But what do I gain from this?" Euclid told his slave to give the boy a coin, so that the student would see an actual profit."
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Mario Livio |
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6715109
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Pythagoras apparently wrote nothing, and yet his influence was so great that the more attentive of his followers formed a secretive society, or brotherhood, and were known as the Pythagoreans. Aristippus of Cyrene tells us in his Account of Natural Philosphers that Pythagoras derived his name from the fact that he was speaking (agoreuein) truth like the God at Delphi (tou Pythiou).
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Mario Livio |
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36c0fb0
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Mind-boggling, isn't it? Centuries before the question of why mathematics was so effective in explaining nature was even asked, Galileo thought he already knew the answer! To him, mathematics was simply the language of the universe. To understand the universe, he argued, one must speak this language. God is indeed a mathematician.
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Mario Livio |
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f976d4a
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You may begin to realize that groups will pop up wherever symmetries exist. In fact, the collection of all the symmetry transformations of any system always from a group.
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Mario Livio |
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536c8de
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To claim that mathematics is purely a human invention and is successful in explaining nature only because of evolution and natural selection ignores some important facts in the nature of mathematics and in the history of theoretical models of the universe. First, while the mathematical rules (e.g., the axioms of geometry or of set theory) are indeed creations of the human mind, once those rules are specified, we lose our freedom. The defini..
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Mario Livio |
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e5da224
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populations typically have such high reproduction potential that if unchecked they would increase exponentially.
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Mario Livio |
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5ae269f
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The environment is the agent that does the shaking of the sieve.
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Mario Livio |
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e186fcd
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Unlike most mathematical discoveries, however, no one was looking for a theory of groups or even a theory of symmetries when the concept was discovered. Quite the contrary; group theory appeared somewhat serendipitously, out of a millenia-long search for a solution to an algebraic equation. Befitting its description as a concept that crystallized simplicity out of chaos, group theory was itself born out of one of the most tumultuous stories..
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Mario Livio |
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c456681
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Leonardo had considerable interest in geometry, especially for its practical applications in mathematics. In his words: "Mechanics is the paradise of the mathematical sciences, because by means of it one comes to the fruits of mathematics."
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Mario Livio |
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6834dca
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Here, however, is where his genius truly took off. Galois managed to associate with each equation a sort of "genetic code" of that equation-the Galois group of the equation-and to demonstrate that the properties of the Galois group determine whether the equation is solvable by a formula or not. Symmetry became the key concept, and the Galois group was a direct measure of the symmetry properties of an equation."
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Mario Livio |
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5938434
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In it, Porphyry says about Pythagoras: "He himself could hear the harmony of the Universe, and understood the music of the spheres, and the stars which move in concert with them, and which we cannot hear because of the limitations of our weak nature."
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Mario Livio |
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d2f9a73
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The beauty of the principle idea of string theory is that all the known elementary particles are supposed to represent merely different vibration modes of the same basic string. Just as a violin or a guitar string can be plucked to produce different harmonics, different vibrational patterns of a basic string correspond to distinct matter particles, such as electrons and quarks. The same applies to the force carriers as well. Messenger parti..
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Mario Livio |
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9dcc170
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Having witnessed in his own life much agony and the horrors of war, Kepler concluded that Earth really created two notes, mi for misery ("miseria" in Latin) and fa for famine ("fames" in Latin). In Kepler's words: "the Earth sings MI FA MI, so that even from the syllable you may guess that in this home of ours Misery and Famine hold sway."
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Mario Livio |
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22b7364
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Wolfram, one of the most innovative thinkers in scientific computing and in the theory of complex systems, has been best known for the development of Mathematica, a computer program/system that allows a range of calculations not accessible before. After ten years of virtual silence, Wolfram is about to emerge with a provocative book that makes the bold claim that he can replace the basic infrastructure of science. In a world used to more th..
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Mario Livio |
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6d0a744
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For example, the central idea in Einstein's theory of general relativity is that gravity is not some mysterious, attractive force that acts across space but rather a manifestation of the geometry of the inextricably linked space and time. Let me explain, using a simple example, how a geometrical property of space could be perceived as an attractive force, such as gravity. Imagine two people who start to travel precisely northward from two d..
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Mario Livio |
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8aa43c4
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Pythagoras is in fact credited with having coined the words "philosophy" ("love of wisdom") and "mathematics" ("that which is learned"). To him, a "philosopher" was someone who "gives himself up to discovering the meaning and purpose of life itself...to uncover the secrets of nature." Pythagoras emphasized the importance of learning above all other activities, because, in his words, "most men and women, by birth or nature, lack the means to..
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Mario Livio |
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87ecc6a
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The Pythagoreans were probably the first to recognize the concept that the basic forces in the universe may be expressed through the language of mathematics.
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Mario Livio |
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Group theory has been called by the noted mathematics scholar James R. Newman "the supreme art of mathematical abstraction." It derives its incredible power from the intellectual flexibility afforded by its definition."
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Mario Livio |
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8268685
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The result that Noether obtained was stunning. She showed that to every continuous symmetry of the laws of physics there corresponds a conservation law and vice versa. In particular, the familiar symmetry of the laws under translations corresponds to conservation of momentum, the symmetry with respect to the passing of time (the fact that the laws do not change with time) gives us conservation of energy, and the symmetry under rotations pro..
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Mario Livio |
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d59a5fd
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In mathematics, if you are of quick mind, you can get to the "frontline" of cutting-edge research very quickly. In some other domains you may have to read entire thick volumes first. Moreover, if you have been for too long in a certain domain, you get conditioned to think like everybody else. When you are new, you are not compelled to the ideas of the people around you. The younger you are, the more likely you are to be truly original."
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Mario Livio |
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124c8af
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So mathematics is indeed extraordinarily effective for some descriptions, especially those dealing with fundamental science, but it cannot describe our universe in all its dimensions. To some extent, scientists have selected what problems to work on based on those problems being amenable to a mathematical treatment.
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Mario Livio |
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79708db
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The number 6 was the first perfect number, and the number of creation. The adjective "perfect" was attached that are precisely equal to the sum of all the smaller numbers that divide into them, as 6=1+2+3. The next such number, incidentally, is 28=1+2+4+7+14, followed by 496=1+2+4+8+16+31+62+124+248; by the time we reach the ninth perfect number, it contains thirty-seven digits. Six is also the product of the first female number, 2, and the..
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Mario Livio |
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34625a7
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In the Middle Ages, the Elements was translated into Arabic three times. The first of these translations was carried out by al-Hajjaj ibn Yusuf ibn Matar, at the request of Caliph Harun ar-Rashid (ruled 786 - 809), who is familiar to us through the stories in The Arabian Nights. The Elements was first made known in Western Europe through Latin translations of Arabic versions. English Benedictine monk Adelard of Bath (ca. 1070 - 1145), who a..
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Mario Livio |
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76d756b
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So, what is light? Is it a pure bombardment by particles (photons) or a pure wave? Really, it is neither. Light is a more complicated physical phenomenon than any single one of these concepts, which are based on classical physical models, can describe. To describe the propagation of light and to understand the phenomena like interference, we can and have to use the electromagnetic wave theory. When we want to discuss the interaction of ligh..
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Mario Livio |
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While Euclid himself may not have been the greatest mathematician who ever lived, he was certainly the greatest teacher of mathematics.
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Mario Livio |
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57a9b45
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What can we conclude from all of these insights in terms of the role of symmetry in the cosmic tapestry? My humble personal summary is that we don't know yet whether symmetry will turn out to be the most fundamental concept in the workings of the universe. Some of the symmetries physicists have discovered or discussed over the years have later been recognized as being accidental or only approximate. Other symmetries, such as general covaria..
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Mario Livio |
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7696bb6
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Tegmark argues that "our universe is not just described by mathematics-it is mathematics" [emphasis added]. His argument starts with the rather uncontroversial assumption that an external physical reality exists that is independent of human beings. He then proceeds to examine what might be the nature of the ultimate theory of such a reality (what physicists refer to as the "theory of everything"). Since this physical world is entirely indep..
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Mario Livio |
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0ef7c28
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The result that Noether obtained was stunning. She showed that to every continuous symmetry of the laws of physics there corresponds a conservation law and vice versa.
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Mario Livio |
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deb8618
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Unfortunately, I do not find Tegmark's line of reasoning to be extremely compelling. The leap from the existence of an external reality (independent of humans) to the conclusion that, in Tegmark's words, "You must believe in what I call the mathematical universe hypothesis: that our physical reality is a mathematical structure," involves, in my opinion, a sleight of hand. When Tegmark attempts to characterize what mathematics really is, he ..
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Mario Livio |
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Noether's theorem fused together symmetries and conservation laws-these two giant pillars of physics are actually nothing but different facets of the same fundamental property.
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Mario Livio |
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Based on these interviews, he compiled a list of ten dimensions of complexity-ten pairs of apparently antithetical characteristics that are often both present in the creative minds. The list includes: 1. Bursts of impulsiveness that punctuate periods of quiet and rest. 2. Being smart yet extremely naive. 3. Large amplitude swings between extreme responsibility and irresponsibility. 4. A rooted sense of reality together with a hefty dose of ..
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Mario Livio |
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bf8fe26
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This development had dramatic philosophical consequences. As in the case of the non-Euclidean geometries in the nineteenth century, there wasn't just one definitive set theory, but rather at least four! One could make different assumptions about infinite sets and end up with mutually exclusive set theories. For instance, once could assume that both the axiom of choice and the continuum hypothesis hold true and obtain one version, or that bo..
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Mario Livio |
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5b7fb25
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This was undoubtedly one of symmetry's greatest success stories. Glashow, Wienberg, and Salam managed to unmask the electromagnetic and weak forces by recognizing that underneath the differences in the strengths of these two forces (the electromagnetic force is about a hundred thousand times stronger within the nucleus) and the different masses of the messenger particles lay a remarkable symmetry. The forces of nature take the same form if ..
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Mario Livio |
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518ac3f
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The biggest stumbling block that has traditionally plagued all the unification endeavors has been the simple fact that on the face of it, general relativity and quantum mechanics really appear to be incomprehensible. Recall that the key concept of quantum theory is the uncertainty principle. When you try to probe positions with an ever-increasing magnification power, the momenta (or speeds) start oscillating violently. Below a certain minus..
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Mario Livio |
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818a6f1
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On the question of whether mathematics was discovered or invented, Pythagoras and the Pythagoreans had no doubt-mathematics was real, immutable, omnipresent, and more sublime than anything that could conceivably emerge from the feeble human mind. The Pythagoreans literally embedded the universe into mathematics. In fact, to the Pythagoreans, God was not a mathematician-mathematics was God! The importance of the Pythagorean philosophy lies n..
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Mario Livio |
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4676420
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Quarks come in six "flavors" that were given the rather arbitrary names: up, down, strange, charm, top, and bottom. Protons, for instance, are made of two up quarks and one down quark, while neutrons consist of two down quarks and one up quark. Other than ordinary electric charge, quarks possess another type of charge, which has been fancifully called color, even though it has nothing to do with anything we can see. In the same way that the..
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Mario Livio |
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6b2ac87
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The forces of nature are color blind. Just as an infinite chessboard would look the same if we interchanged black and white, the force between a green quark and a red quark is the same as that between two blue quarks, or a blue quark and a green quark. Even if we were to use our quantum mechanical "palette" and replace each of the "pure" color states with a mixed-color state (e.g., "yellow" representing a mixture of red and green or "cyan" ..
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Mario Livio |
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ea020db
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The realization that symmetry is the key to the understanding of the properties of subatomic particles led to an inevitable question: Is there an efficient way to characterize all of these symmetries of the laws of nature? Or, more specifically, what is the basic theory of transformations that can continuously change one mixture of particles into another and produce the observed families? By now you have probably guessed the answer. The pro..
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Mario Livio |
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1ceecda
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In his book Opticks, Newton made it clear that he did not believe that the laws of nature by themselves were sufficient to explain the universe's existence-God was the creator and sustainer of all the atoms that make up the cosmic matter: "For it became him [God] who created them [the atoms] to set them in order. And if he did so, it's unphilosophical to seek for any other Origin of the World, or to pretend that it might arise out of a Chao..
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Mario Livio |
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Moreover, Galileo argued that by pursuing science using the language of mechanical equilibrium and mathematics, humans could understand the divine mind.
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Mario Livio |
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Unlike classically spinning bodies, such as tops, however, where the spin rate can assume any value fast or slow, electrons always have only one fixed spin. In the units in which this spin is measured quantum mechanically (called Planck's constant) the electrons have half a unit, or they are "spin-1/2" particles. In fact, all the matter particles in the standard model-electrons, quarks, neutrinos, and two other types called muons and taus-a..
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Mario Livio |