Why Beauty Is Truth: A History of Symmetry

This is an excellent book, although to fully understand it you need some good background in math and physics. It traces 4000 years of research in mathematics and physics, from Babylonic science (to whom we owe the sexagesimal system) to Ed Witten and superstrings. The thread of the story is symmetry, a concept that leads to group theory via the efforts to solve some the antiquity's problems (for example, the duplication of the cube) and the polynomial equations, specially the quintic. Although I am an avid reader of this kind of books I learnt quite a few things and others, although not new to me, I found were very well explained. 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.

This book made math and its history extremely readable. Its core idea was symmetry and how it acted as the driving force behind many mathematical inspirations. Ian Stewart is a master writer and he proves himself again in this book. He defines symmetry not untill p.118, where he sees symmetry as a kind of "transformation" which when applied to a mathematical object preserves its structure. Then he explains these individual aspects of symmetry in relation to Galois' groups. Near the end of the book, he brought physics into the discussion, and showed how deep abstract sense of beauty also played a crucial role in developing physical ideas. To some, it may appear bizarre, as most of the book talks about mathematicians and their 'beauties,' and suddenly physics creeps in. But in hindsight, the sense of beauty and truth is never complete without the taste of reality. Physics serves that purpose. And so he ends: "In physics, beauty does not automatically ensure truth, but it helps. In mathematics, beauty MUST be true - beacause anything false is ugly." A true ending to a beautiful book.

Some of the reviews of this book seem to feel it doesn't present enough group theory. I think they are looking for a more technical book than Stewart meant to write, and so they are downgrading the book for reasons that are not fair to the book. I reviewed a book by Mario Livio called "The Equation that Couldn't Be Solved," and gave it 5 stars. After reading this book, I almost want to go back and lower my rating of Livio's book, but of course, I shouldn't do that just because a better book has come out since. Livio's book concentrates on a shorter timespan than this, but both feature the same things -- mathematicians' attempts to solve equations of higher and higher degrees, from quadratics to cubics to quartics, and failure to find a solution to the quintic, only to find (due to the work of Abel and Galois) that it couldn't be done; and Galois' invention of group theory to make his proof, followed by other mathematicians' revelation that group theory is just what the doctor ordered to explain symmetry. Stewart's book goes further back in time than Livio's, and also devotes more space to the modern uses of symmetry in physics. So it puts everything in more context. And, simply put, Stewart is a captivating writer. I enjoyed Livio's book, but I could hardly put down Stewart's. This book gets a high 5-star rating from me. But it IS a book for the non-specialist. It isn't a course in group theory, or the Galois theory of equations; it is an attempt to give a non-mathematician some idea of these subjects. It should not be rated on a set of criteria that ignore what Stewart was trying to do. The negative comments really are unjustified; but yes, I'll warn you away from this if you expect it to teach you all the group theory you'll need to do particle physics, or crystallography, or any of the subjects that depend on group theoretic concepts of symmetry these days.

"Beauty" in Stewart's title refers to symmetry in mathematics and physics, and to the mathematical structures called groups, which express this symmetry. "Truth" refers to the fact that the fundamental laws of the universe are described by such symmetries. Before Stewart goes into this, he builds up for about 100 pages, giving the historical background of the ideas leavened with some biographical sketches. Then he gives two simple examples which form a basis for going into the later topics. I can't match Stewart's simplicity in a brief review, but I hope I can give you an idea of the nature of the examples. Symmetry here has a somewhat more general meaning than in ordinary language. Ordinarily we say that something is symmetric if it looks the same as its mirror reflection. It is often said that a starfish has "radial symmetry" because, if it is rotated by 72 degrees (1/5 of a circle), it still looks the same, right down to the legs pointing in the same directions. Stewart considers the rotations and reflections of an equilateral triangle and defines a sort of "multiplication" of these turnings. The turnings together with the "multiplication" have a structure known as a "group". (It is called "multiplication" because it follows the same rules as multiplication of numbers. Any set of things which follow these rules is a group.) There is also purely mathematical symmetry. For example, suppose you have a formula containing 3 numbers. If you rearrange those numbers in any order and the value of the formula is still the same, that rearrangement is called a symmetry. Instead of preserving the shape of an object, it preserves the value of an expression. Stewart shows that there is a deep connection between this group and the triangle group: both have the same multiplication table. From there, Stewart goes on to applications of groups, symmetry and connections, mostly in physics. Here, he can't go into as much detail because the mathematics is too advanced. Like others who write on Physics for a general audience, he gives an impression of what the physics is like. This is why I called it a "walking tour". Unlike many others, however, he makes it clear he's not telling the whole story. For example, when talking about the spin of a particle, authors often have a drawing of a ball with a curved arrow indicating a spinning motion. "The particles did not spin in space, like the Earth or a spinning top. They "spin" -- whatever that means -- in more exotic dimensions." Before I read this, I wasted a lot of time trying to figure out explanations while visualizing a spinning ball. Now I just understand that spin is an abstract property and I have a better feel for the character of the science. I think that many readers will have a clearer notion of Einstein's (and Riemann's) curved space than can be gotten from the misleading "rubber sheet geometry" analogy that is so popular with science writers. As he gets into the physics, Stewart b