Customer Reviews

Why Beauty Is Truth: The History of Symmetry

5 star:		(11)
4 star:		(7)
3 star:		(5)
2 star:		(1)
1 star:		(1)

 

 

"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 brings up a new type of mathematical object, the Lie groups. I have seen these a few times before with no understanding at all. I assumed that they involved some abstruse math that would require more work than I was willing to put out. But Stewart defines two of these groups, called O(2) and SO(2), and they turn out to be very like the triangle group. No one had been able to explain this for general readers before because no one was prepared to spend over a hundred pages working up to it.

There is a lot of good material in this book and none of it requires any knowledge of high school math, although a little algebra will enhance some people's appreciation.

At this point I have to mention that I have a Ph.D. in math, although not in areas related to group theory. Much of the material in this book is new to me. Over the past few decades I have spent considerable energy learning how the general public sees math and science and thinking of how to explain ideas in non-technical ways, so I am confidant when I say: Why Beauty Is Truth is an excellent book to give general readers a view of how the beauty of symmetry, expressed in the language of groups, has helped to shape modern physics.

Addendum: (This is strictly for people who want to think seriously about the math.) The "multiplication" I mentioned in the triangle example means one turning followed by another. Once you get to the definition in the book, you might like to do some calculations to verify that the turnings really do form a group, that the "multiplication" table is correct, and that the triangle group and the permutation group have the same table. (This kind of equivalence is very important in mathematics.) I don't recommend this for all readers, but for some it will give a real insight into how mathematicians work. I do recommend it very strongly for young readers who might like to major in math.

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.

I have always enjoyed Professor Ian Stewart's works for general audiences, including "Letters to a Young Mathematician" and "Flatterland", among others. In "Why Beauty Is Truth: A History of Symmetry", Stewart continues to explain seemingly esoteric and difficult mathematical topics with a clarity and humanity that illuminate not only the topics themselves, but also the people who developed them and the importance of their work to us in the present day.

In his latest book, "Why Beauty Is Truth", Stewart recounts the history of a concept most of us understand intuitively, symmetry, by describing the lives of people who made important contributions to the mathematics of this seemingly simply concept which turned out to have extraordinary implications. From the development of ancient number systems and algebra to the discovery of Lie groups, Stewart explains the mathematics and concepts in an intuitive way, sprinkling in equations when necessary, but mostly relying on his ability to imagine how a non-mathematician might best understand even the most abstract concepts, whether by example, metaphor, or even some fictional drama.

Stewart is the rare mathematician who seems equally at home with the technical aspects of his subject and its history, including the biographies of those who made important contributions. Stewart is also a fine writer and enthusiastic popularizer, showing how the development of symmetry from the beginnings of counting has led to some of the most important developments in physics, including general relativity and string theory. Math and physics enthusiasts will undoubtedly enjoy "Why Beauty Is Truth", as will the curious lay reader who enjoys new discoveries and lively, engaging and intelligent writing.

I actually do agree with the terse reviewer below who mentions there is not enough on group theory. I think the reader or buyer should be forewarned that there is a lot of historical and biographical information in here, going back to the ancient greeks: Pythagoras, etc., traveling through history until string theory (on which subject the writer seems to be quite enthusiastic albeit fence-sitting). Throughout that excursion there is a lot of biographical information, some of it quite uninteresting and irrelevant (in my opinion!). Certainly some people might be interested to read the lives of the mathematicians but I was hoping for a book that dallied with the more philosophical implications of beauty and truth, mathematics and reality, such as for example Paul Davies does so well. As such I don't think he really explains why it might be correct to say that "beauty is truth." Nonetheless the book is really really well written and approachable, and Ian Stewart does a fantastic job of explaining complicated math concepts. Towards the end it feels like he is hurrying through some of the most interesting topics, such as how group theory applies to the standard model of quantum mechanics, which seems to be the most surprising or fascinating application of the concept of symmetry to reality. A few physicists get bunched together in the last couple of chapters where they might each have merited a chapter on their own.

This book covers an enormous range of topics beginning with Mesopotamia number systems and ending with string theory. It simultaneously describes mathematical theories, the history of how these ideas evolved over time, and details about the lives of the mathematicians. Several of the brief biographies are very well done; the treatments of Gauss, Omar Khayyám, and Galois are outstanding. Others are sketchy, hardly more than a list of parent's occupations, siblings, spouse, and children. As a result of the broad coverage, each mathematical concept gets very brief treatment. I often felt that I wasn't given enough information to understand a concept. Lie groups, in particular, turn out to be very important for contemporary physics but the description is so brief and jargon encrusted that the physical applications were unintelligible to me.
The author is not certain about his intended audience. He apologizes to the reader for the complexity of the solution to the cubic equation, even though this is a straightforward extension of high school algebra. Yet later on he assumes that the reader will easily grasp that a Fano plane is a finite projective geometry. The book was simultaneously too nontechnical and too technical for me (a computer technologist and a former scientist).
It is not clear what the purpose of the book is. Many of the topics covered have no obvious connection to symmetry except in the sense that everything is related to symmetry. The historical evolution of representations of numbers is interesting, for example, but doesn't help understanding the multidimensional algebras that somehow relate to 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.

Ian Stewart's WHY BEAUTY IS TRUTH: A HISTORY OF SYMMETRY discusses the history and nature of the concept of symmetry, from early times to the latest physics involvement. This analysis comes from a famous mathematician who charts the discoveries and investigative processes of fellow scientists and geniuses as they developed the concept of symmetry and drew important connections between its science and nature. Any collection strong in science needs this discussion.

I agree completely with the review by Israel Ramirez. Stewart religiously avoids mathematical expressions throughout, assuming his audience will choke on anything more than a polynomial equation, but he doesn't think twice about spewing esoteric math and physics jargon when attempting to explain fantastically complicated concepts in words. To his credit, it works some of the time, but any honest reader will admit it doesn't work much of the time. As others have noted, there are many positive things about the book, but by the end the author is throwing new thoughts into the mix helter skelter as if cleaning out his ideas closet, and it all just falls apart. What are we supposed to take away from nonsense such as: "So now the general opinion is that the exceptional Lie groups exist because of the wisdom of the deity in permitting the octonions to exist."??

Why Beauty Is Truth: A History of Symmetry
S. Marsh statement "you can trisect an angle" is not true in its historical context.

Historical context: It is not possible to trisect all angles using only a compass and a straightedge (unmarked ruler).

In his book, Stewart says that it is possible to compute values to great precision,(which includes using iteration) but not by compass and ruler. He does mention that it is possible to trisect some angles, specifically mentioning 180 which trisects to 60 which can be constructed by making a regular hexagon. But trisecting 60 degrees by compass and ruler to produce 20 degrees is impossible, Note that 20 is the exact value of a trisected 60 degree angle but you cannot construct that angle, with a straightedge and compass.

As Stewart makes clear in this book, the important thing is not that you can't, but why you can't. And the why leads to group theory and other advances.

I found this book to be extremely interesting. Group theory is new to me. I found this book is an introduction as to why it was important to Einstein and to modern physics.

I recommend this book.

I found the following on-line tutorial on Galois theory useful:
http://nrich.maths.org/public/viewer.php?obj_id=1422