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Author: Mario Livio
Quotes of Author: Mario Livio

Mario Livio _ Is God a Mathematician?
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 concepts. In some cases, the proofs generated the theorems-mathematicians examined what they could prove and from that they deduced the theorems. In others, as described by Archimedes in The Method, they first found the answer to a particular question they were interested in, and then they worked out the proof.Typically, the concepts were inventions. Prime numbers as a concept were an invention, but all the theorems about prime numbers were discoveries. The mathematicians of ancient Babylon, Egypt, and China never invented the concept of prime numbers, in spite of their advanced mathematics. Could we say instead that they just did not "discover" prime numbers? Not any more than we could say that the United Kingdom did not "discover" a single, codified, documentary constitution. Just as a country can survive without a constitution, elaborate mathematics could develop without the concept of prime numbers. And it did!Do we know why the Greeks invented such concepts as the axioms and prime numbers? We cannot be sure, but we could guess that this was part of their relentless efforts to investigate the most fundamental constituents of the universe. Prime numbers were the basic building blocks of matter. Similarly, the axioms were the fountain from which all geometrical truths were supposed to flow. The dodecahedron represented the entire cosmos and the golden ratio was the concept that brought that symbol into existence.
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Mario Livio _ The Equation That Couldn't Be
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 we do that, however, the answer to the question, Can we find translation-symmetric music? is a resounding yes. As Russian crystal physicist G. V. Wulff wrote in 1908: "The spirit of music is rhythm. It consists of the regular, periodic repetition of parts of the musical composition...the regular repetition of identical parts in the whole constitutes the essence of symmetry." Indeed, the recurring themes that are so common in musical composition are the temporal equivalents of Morris's designs and symmetry under translation. Even more generally, compositions are often based on a fundamental motif introduced at the beginning and then undergoing various metamorphoses.
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Mario Livio _ The Equation That Couldn't Be
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, Birkhoff's theory heralds symmetry as a crucial aesthetic element.
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Mario Livio _ The Equation That Couldn't Be
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.
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Mario Livio _ The Equation That Couldn't Be
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. Addition, for instance, is associative: {5 + 7} + 13 = 25 and 5 + {7 + 13} = 25, where the parentheses, the "punctuation marks" of mathematics, indicate which pair you add first.3. Identity element. The group has to contain an identity element such that when combined with any member, it leaves the member unchanged. In the group of integers, the identity element is the number zero. For example, 0 + 3 = 3 + 0 = 3.4. Inverse. For every member in the group there must exist an inverse. When a member is combined with its inverse, it gives the identity element. For the integers, the inverse of any number is the number of the same absolute value, but with the opposite sign: e.g., the inverse of 4 is -4 and the inverse of -4 is 4; 4 + {-4} = 0 and {-4} + 4 = 0.The fact that this simple definition can lead to a theory that embraces and unifies all the symmetries of our world continues to amaze even mathematicians.
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Mario Livio _ The Equation That Couldn't Be
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 "charges" of the three quarks that make up a proton or a neutron are all different {red, blue, green}, and they add up to give zero color charge or "white" {equivalent to being electrically neutral in electromagnetism}. Since color symmetry is at the base of the gluon-mediated force between quarks, the theory of these forces has become known as quantum chromodynamics. The marriage of the electroweak theory {which describes the electromagnetic and weak forces} with quantum chromodynamics {which describes the strong force} produced the standard model-the basic theory of elementary particles and the physical laws that govern them.
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Mario Livio _ The Golden Ratio: The Story of
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.
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Mario Livio _ The Equation That Couldn't Be
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 through the discovery of the most suitable Lie group that contains the product U{1} X SU{2} x SU{3}.
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Mario Livio _ The Equation That Couldn't Be
Is it odd how asymmetricalIs "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.
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Mario Livio _ The Golden Ratio: The Story of
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. This effect was explained by Einstein in 1905 as resulting from the collisions that the colloidal particles experience with the molecules of the surrounding fluid. Each single collision has a negligible effect, because the pollen grains are millions of times more massive than the water molecules, but the persistent bombardment has a cumulative effect. Amazingly, the same model was found to apply to the motions of stars in star clusters. There the Brownian motion is produced by the cumulative effect of many stars passing by any given star, with each passage altering the motion {through gravitational interaction} by a tiny amount.
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Mario Livio _ The Equation That Couldn't Be
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 into the outlet directly. The mathematician, on the other hand, says that he would take the iron to the room without the outlet, since that problem has already been solved.
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Mario Livio _ The Equation That Couldn't Be
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 exchange between them bundles of energy called photons. The photon is therefore the messenger of electromagnetism. The electroweak theory predicted the existence of close siblings to the photon, which play the messenger role for the weak force. These never-before-seen particles were prefigured to be about ninety times more massive than the proton and to come in both an electrically charged {called W} and a neutral {called Z} variety. Experiments performed at the European consortium for nuclear research in Geneva {known as CERN for Conseil Europeen pour la Recherche Nucleaire} discovered the W and Z particles in 1983 and 1984 respectively.
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