Mario Livio > Quotes

“For Newton, the world's very existence and the mathematical regularity of the observed cosmos were evidence for God's presence.”
― Mario Livio, Is God a Mathematician?

“Our mathematics is the symbolic counterpart of the universe we perceive, and its power has been continuously enhanced by human exploration.”
― Mario Livio, The Golden Ratio: The Story of Phi, the World's Most Astonishing Number

“The key point to keep in mind, however, is that symmetry is one of the most important tools in deciphering nature's design.”
― Mario Livio, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

“Because of the "divine" properties attributed to the Golden Ratio, mathematician Clifford A. Pickover suggested that we should refer to that point as "the Eye of God.”
― Mario Livio, The Golden Ratio: The Story of Phi, the World's Most Astonishing Number

“we are much better at judging other peoplethan at analyzing ourselves.”
― Mario Livio, Brilliant Blunders: From Darwin to Einstein - Colossal Mistakes by Great Scientists That Changed Our Understanding of Life and the Universe

“Even today, I am in total awe of the following wondrous chain of ideas and interconnections. Guided throughout by principles of symmetry, Einstein first showed that acceleration and gravity are really two sides of the same coin. He then expanded the concept to demonstrate that gravity merely reflects the geometry of spacetime. The instruments he used to develop the theory were Riemann's non-Euclidean geometries-precisely the same geometries used by Felix Klein to show that geometry is in fact a manifestation of group theory (because every geometry is defined by its symmetries-the objects it leaves unchanged). Isn't this amazing?”
― Mario Livio, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

“Put simply, the cosmological argument claims that since the physical world had to come into existence somehow, there must be a First Cause, namely, a creator God.”
― Mario Livio, Is God a Mathematician?

“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 first masculine number, 3. The Hellenistic Jewish philosopher Philo Judaeus of Alexandria (ca. 20 B.C.-c.a. A.D. 40), whose work brought together Greek philosophy and Hebrew scriptures, suggested that God created the world in six days because six was a perfect number. The same idea was elaborated upon by St. Augustine (354-430) in The City of God: "Six is a number perfect in itself, and not because God created the world in six days; rather the contrary is true: God created the world in six days because this number is perfect, and it would remain perfect, even if the work of the six days did not exist." Some commentators of the Bible regarded 28 also as a basic number of the Supreme Architect, pointing to the 28 days of the lunar cycle. The fascination with perfect numbers penetrated even into Judaism, and their study was advocated in the twelfth century by Rabbi Yosef ben Yehudah Ankin in his book, Healing of the Souls.”
― Mario Livio, The Golden Ratio: The Story of Phi, the World's Most Astonishing Number

“The Golden Rectangle is the only rectangle with the property that cutting a square from it produces a similar rectangle.”
― Mario Livio, The Golden Ratio: The Story of Phi, the World's Most Astonishing Number

“Pythagoras was born around 570 B.C. in the island of Samos in the Aegean Sea (off Asia Minor), and he emigrated sometime between 530 and 510 to Croton in the Dorian colony in southern Italy (then known as Magna Graecia). Pythagoras apparently left Samos to escape the stifling tyranny of Polycrates (died ca. 522 B.C.), who established Samian naval supremacy in the Aegean Sea. Perhaps following the advice of his presumed teacher, the mathematician Thales of Miletus, Pythagoras probably lived for some time (as long as twenty-two years, according to some accounts) in Egypt, where he would have learned mathematics, philosophy, and religious themes from the Egyptian priests. After Egypt was overwhelmed by Persian armies, Pythagoras may have been taken to Babylon, together with members of the Egyptian priesthood. There he would have encountered the Mesopotamian mathematical lore. Nevertheless, the Egyptian and Babylonian mathematics would prove insufficient for Pythagoras' inquisitive mind. To both of these peoples, mathematics provided practical tools in the form of "recipes" designed for specific calculations. Pythagoras, on the other hand, was one of the first to grasp numbers as abstract entities that exist in their own right.”
― Mario Livio, The Golden Ratio: The Story of Phi, the World's Most Astonishing Number

“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 Chaos by the mere Laws of Nature.” In other words, to Newton, God was a mathematician (among other things), not just as a figure of speech, but almost literally-the Creator God brought into existence a physical world that was governed by mathematical laws.”
― Mario Livio, Is God a Mathematician?

“Twentieth-century British mathematician G.H. Hardy also believed that the human function is to "discover or observe" mathematics rather than to invent it. In other words, the abstract landscape of mathematics was there, waiting for mathematical explorers to reveal it.”
― Mario Livio, The Golden Ratio: The Story of Phi, the World's Most Astonishing Number

“Surprisingly, palindromes appear not just in witty word games but also in the structure of the male-defining Y chromosome. The Y's full genome sequencing was completed only in 2003. This was the crowning achievement of a heroic effort, and it revealed that the powers of preservation of this sex chromosome have been grossly underestimated. Other human chromosome pairs fight damaging mutations by swapping genes. Because the Y lacks a partner, genome biologists had previously estimated that its cargo was about to dwindle away in perhaps as little as five million years. To their amazement, however, the researchers on the sequencing team discovered that the chromosome fights withering with palindromes. About six million of its fifty million DNA letters form palindromic sequences-sequences that read the same forward and backward on the two strands of the double helix. These copies not only provide backups in case of bad mutations, but also allow the chromosome, to some extent, to have sex with itself-arms can swap position and genes are shuffled. As team leader David Page of MIT has put it, "The Y chromosome is a hall of mirrors.”
― Mario Livio, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

“Indeed, the quality that made Newton's theories truly stand out-the inherent characteristic that turned them into inevitable laws of nature-was precisely the fact that they were all expressed as crystal-clear, self-consistent mathematical relations.”
― Mario Livio, Is God a Mathematician?

“The Golden Ratio has the unique properties that we produce its square by simply adding the number 1 and its reciprocal by subtracting the number 1.”
― Mario Livio, The Golden Ratio: The Story of Phi, the World's Most Astonishing Number

“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.”
― Mario Livio, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

“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).”
― Mario Livio, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

“Some ancient Indian texts claim that numbers are almost divine, or "Brahma-natured.”
― Mario Livio, The Golden Ratio: The Story of Phi, the World's Most Astonishing Number

“Imagine what would have happened had the logicist endeavor been entirely successful. This would have implied that mathematics stems fully from logic-literally from the laws of thought. But how could such a deductive science so marvelously fit natural phenomena? What is the relation between formal logic (maybe we should even say human formal logic) and the cosmos? The answer did not become any clearer after Hilbert and Godel. Now all that existed was an incomplete formal "game," expressed in mathematical language. How could models based on such an "unreliable" system produce deep insights about the universe and its workings?”
― Mario Livio, Is God a Mathematician?

“The most famous Diophantine equation in history is the one known as Fermat's last Theorem, the celebrated statement by Pierre de Fermat (1601-55) that there are no whole number solutions to the equation x^n + y^n = z^n, where n is any number greater than 2. When n = 2, there are many solutions (in fact an infinite number). For instance, 3^2 + 4^2 = 5^2 (9 + 16 = 25); or 12^2 +5^2 = 13^2 (144 + 25 = 169). Miraculously, when we go from n = 2 to n = 3, there are no whole numbers x,y,z that satisfy x^3 + y^3 = z^3, and the same is true for any other value of n that is greater than 2. Appropriately, it was in the margin of the second book of Diophantus's Arithmetica, which Fermat was eagerly reading, that he wrote his extraordinary claim-one that took no fewer than 356 years to prove.”
― Mario Livio

“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 according to some stories was traveling in Spain disguised as a Muslim student, got hold of an Arabic text and completed the translation into Latin around 1120. This translation became the basis of all editions in Europe until the sixteenth century. Translations into modern languages followed.”
― Mario Livio, The Golden Ratio: The Story of Phi, the World's Most Astonishing Number

“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 independent of humans, Tegmark maintains, its description must be free of any human "baggage" (e.g., human language, in particular). In other words, the final theory cannot include any concepts such as "subatomic particles," "vibrating strings," "warped spacetime," or other humanly conceived constructs. From this presumed insight, Tegmark concludes that the only possible description of the cosmos is one that involves only abstract concepts and the relations among them, which he takes to be the working definition of mathematics.”
― Mario Livio, Is God a Mathematician?

“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 in the history of mathematics. Almost four thousand years of intellectual curiosity and struggle, spiced with intrigue, misery, and persecution, culminated in the creation of the theory in the nineteenth century. This amazing story, chronicled in the next three chapters, began with the dawn of mathematics on the banks of the Nile and Euphrates rivers.”
― Mario Livio, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

“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 fantasy and imagination.

5. Alternating periods of introversion and extroversion.

6. Being simultaneously humble and proud.

7. Psychological androgyny-no clear adherence to gender role stereotyping.

8. Being rebellious and iconoclastic yet respectful to the domain of expertise and its history.

9. Being on one had passionate but on the other objective about one's own work.

10. Experiencing suffering and pain mingled with exhilaration and enjoyment.”
― Mario Livio, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

“Galileo established what has since become the modern approach to the study of all natural phenomena.”
― Mario Livio, Galileo: And the Science Deniers

“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.”
― Mario Livio, The Golden Ratio: The Story of Phi, the World's Most Astonishing Number

“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.”
― Mario Livio, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

“While composite faces tend, by construction, to also be more symmetric, Langlois found that even after the effects of symmetry have been controlled, averageness was still judged to be attractive. These findings argue for a certain level of prototyping in the mind, since averageness might well be coupled with a prototypical template.”
― Mario Livio, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

“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, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

“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 minuscule length known as the Planck length, the entire tenet of a smooth spacetime is lost. This length (equal to 0.000...4 of an inch, where the 4 is at the thirty-fourth decimal place) determines the scale at which gravity has to be treated quantum mechanically. For smaller scales, space turns into an ever-fluctuating "quantum foam." But the very basic premise of general relativity has been the existence of a gently curved spacetime. In other words, the central ideas of general relativity and quantum mechanics clash irreconcilably when it comes to extremely small scales.”
― Mario Livio, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry