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Why Beauty Is Truth: The History of Symmetry Hardcover – April 10, 2007
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From Publishers Weekly
Anyone who thinks math is dull will be delightfully surprised by this history of the concept of symmetry. Stewart, a professor of mathematics at the University of Warwick (Does God Play Dice?), presents a time line of discovery that begins in ancient Babylon and travels forward to today's cutting-edge theoretical physics. He defines basic symmetry as a transformation, "a way to move an object" that leaves the object essentially unchanged in appearance. And while the math behind symmetry is important, the heart of this history lies in its characters, from a hypothetical Babylonian scribe with a serious case of math anxiety, through Évariste Galois (inventor of "group theory"), killed at 21 in a duel, and William Hamilton, whose eureka moment came in "a flash of intuition that caused him to vandalize a bridge," to Albert Einstein and the quantum physicists who used group theory and symmetry to describe the universe. Stewart does use equations, but nothing too scary; a suggested reading list is offered for more rigorous details. Stewart does a fine job of balancing history and mathematical theory in a book as easy to enjoy as it is to understand.Line drawings. (Apr.)
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*Starred Review* Werner Heisenberg recognized the numerical harmonies at the heart of the universe: "I am strongly attracted by the simplicity and beauty of the mathematical schemes which nature presents us." An accomplished mathematician, Stewart here delves into these harmonies as he explores the way that the search for symmetry has revolutionized science. Beginning with the early struggles of the Babylonians to solve quadratics, Stewart guides his readers through the often-tangled history of symmetry, illuminating for nonspecialists how a concept easily recognized in geometry acquired new meanings in algebra. Embedded in a narrative that piquantly contrasts the clean elegance of mathematical theory with the messy lives of gambling, cheating, and dueling mathematicians, the principles of symmetry emerge in radiant clarity. Readers contemplate in particular how the daunting algebra of quintics finally opened a conceptual door for Evaniste Galois, the French genius who laid the foundations for group theory, so empowering scientists with a new calculus of symmetry. Readers will marvel at how much this calculus has done to advance research in quantum mechanics, relativity, and cosmology, even inspiring hope that the supersymmetries of string theory will combine all of astrophysics into one elegant paradigm. An exciting foray for any armchair physicist! Bryce Christensen
Copyright © American Library Association. All rights reserved
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The journey begins way back in Babylonia where they began to understand how to solve equations. From there we travel forward to Euclid and his "Elements of Geometry" and the discovery of the concepts of proofs and axioms. Algebra arrives on the scene around 830 CE when the developments moved from the Greek world to the Arabic. In that year, Mohamed al-Khwārizmī wrote a book called "al-Jabr" from which comes our word "algebra." There are contributions from others, such as Omar Khayyam (cubic equations), and the Greek mathematician Menaechmus (conic sections). In the 16th century, Girolamo Cardano wrote a book subtitled "The Rules of Algebra," where he assembled methods for solving not only quadratic equations, but also cubic and quartic equations using pure algebra. By the 18th century, Carl Friedrich Gauss developed what is now called the Fundamental Theorem of Algebra. His worked was followed up by his student Georg Bernhard Riemann who generalized Gauss's work on multidimensional spaces creating, in effect, a theory of curved multidimensional spaces - a concept that would later proved crucial in Einstein's work on gravity. We also get an introduction to the contributions of other notables such as Joseph-Louis Lagrange, Paolo Ruffini, and Hans Mathias Abel.
We learn of the gifted Evariste Galois and his contributions to group theory. Steward notes that in Galois's hands mathematics ceased to be a study of numbers and shapes or arithmetic and geometry but became a study of structure. The study of "things" became a study of processes - this followed on the work of Lagrange, Ruffini, and Abel. Galois is the first to "appreciate that mathematical questions could be sometimes best understood by transporting them into a more abstract realm of thought." We learn that a symmetry of an object is some transformation that preserves the object's structure, and Galois's symmetries were permutations of the roots of a an equation. Incidentally, group theoretic methods eventually came to dominate quantum mechanics in the 20th century, "because the influence of symmetry is all-pervasive."
In the 1800s, William Hamilton, almost three hundred years after Cardano indicated that imaginary numbers might be useful, removed the geometric element and reduced complex numbers to pure algebra, and he discovered a new type of numbers called quaternions. In that same century Marius Sophus Lie discovers Lie groups and Lie algebras. Steward notes that these theories "have pervaded almost every branch of mathematics [...] Symmetry is deeply involved in every area of mathematics, and it underlies most of the basic ideas of mathematical physics." Wilhelm Karl Joseph Killing, who Steward calls the "greatest mathematician who ever lived," expanded on Lie's work.
In the latter 19th century, the work of Faraday, Maxwell, Hertz, and Marconi lead the way to the work of one of the greatest minds yet - Albert Einstein. The story gets more intriguing with the discovery of octonions (another type of algebra) in 1843. It was realized that the octonions were the source of the most "bizarre algebraic structures know to mathematics. They explain where Killing's five exceptional Lie groups [...] really come from," and the one group "shows up twice in the symmetry group that forms the basis of 10-dimensional string theory." String theory is one that is prominent in the quest for a "theory of everything" today. Steward then notes that this "opens up an intriguing philosophical possibility: the underlying structure of our universe, which we know to be very special, is singled out by its relationship to a unique mathematical object: the octonions."
The interesting thing is that the potential of many of these mathematical concepts was not know or appreciated at the time of discovery. Yet today, we can look back and see the beauty of it all, and how it all seems to fit together. Steward notes that symmetry "is fundamental to today's scientific understanding of the universe and its origin." It pervades everything from the world of quantum physics to Einstein's concept of relativity.
I have read many books on symmetry and this one provides in my judgment one of the best entry points. Ian Stewart has an engaging style that builds momentum as you go along.
The blue butterfly on the cover is now a very opportune symbol of symmetry. The 2014 Fields Medal (equivalent to the Nobel prize) was given to Artur Avila for his work on nonlinear dynamical systems of which the Lorenz model is prototypical. This model is usually symbolized by a butterfly. In Oriental cultures the butterfly is a symbol of transformation. Symmetry is defined mathematically as invariance under transformations.
Phil Anderson, a Nobel laureate in physics, famously wrote "it is only slightly overstating the case to say that physics is the study of symmetry." If you want to know why and, as importantly, if you want to know how symmetry can be a key concept for the social sciences and even your own life, you should read this book.
Among the first group, the cubic geometric solutions of Persian Omar in the 11th century, the name of Killing (the mathematician who classified simple Lie algebras in one of the most beautiful math papers, according to Stewart), the fact that Liouville rescued Galois papers from oblivion, the relation of octonions to string theory, Hamilton's carving of the fundamental relations of his quaternions in the Broome Bridge, the role of the exceptional Lie groups in physics, Witten's starting career as political journalist, etc.
Among the second: the description of gauge symmetries, the comparison between the unity of life and the unity of the fundamental forces, etc.
The reader will enjoy the well known story of how mathematicians were forced to use complex numbers in trying to apply the cubic formula and the fascinating life of Galois who so unhappily was killed in a duel at the age of 21, a duel that he had apparently exactly 50% chance of survival.
Stewart is critical of the anthropic principle, even in its weak form. According to him a sufficient condition should not be confused with a necessary condition and who knows in which exotic forms can complexity emerge. I think that we also should reflect on his suggestion that the search of a Theory of Everything is a residue of our monotheistic culture.
One of the main themes of the book is the unreasonable effectiveness of mathematics (a famous article by Wigner has this title) and the ethernal dilemma: is mathematics invented or discovered? The exceptional Lie groups seem to be put there by a deity. These are fascinating subjects and no definitive answers can be given.
One little criticism: Stewart does not distinguish properly hadrons and leptons and leds the uneducated reader to believe that all particles are either made of quarks or are gluons.
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