Why Beauty is Truth
The first equations
Although symmetry seems to be predominantly a geometrical property, this book shows that it really gained its importance in mathematics and physics via a different route - that of the solutions of polynomial equations. Stewart starts at the time of the Babylonians, who were able to solve quadratic equations. He shows how the geometry of Euclid can be linked to the solubility of equations and goes on to describe how people gradually worked out how to solve ever more complicated equations. Omar Khayyám, of Rubiayat fame, played an important part in classifying the solution of cubic equations, but it was the sixteenth century before the general solution was finally worked out. We find out about the disagreements which could arise when equation solving competitions meant that knowledge of such solutions could be highly profitable. The general solution to the quartic equation followed soon after, and it must have seemed that the quintic wouldn't be long.
The insolubility of the quintic
By the end of the eighteenth century the quintic still had no general solution by radicals - that is as an expression involving nth roots of quantities. Mathematicians were beginning to put forward proofs of the impossibility of such a solution, but it was the proof of Niels Abel which was finally accepted, although he died young and had scant recognition in his lifetime. This brings us to Évariste Galois, the darling of anyone presenting mathematics to a general audience. Galois was a revolutionary who was killed in a duel at the age of 20, but still managed to revolutionise mathematics with his methods of showing when an equation was soluble by radicals. I felt that Stewart's treatment of Galois' work is the best I have seen for the novice, explaining what is going on with a minimum of technicality. Stewart also explains one of the reasons Galois had problems in getting his work published - people were looking for conditions for solubitity in terms of the cooeficients of a polynomial equation, but the best that could be done was a condition in terms its roots. Galois' work was the beginning of group theory, and the rest of the book shows how this had much influence in later mathematics and physics. Stewart also looks at related proofs such as such as the impossibility of the trisection of an angle with ruler and compasses, and the proof that π and e aren't the roots of integer polynomials
Symmetry and Physics
Stewart goes on to describe the work of William Hamilton, who is best known today for Hamiltonian methods which provided a new way of looking at problems in physics. Hamilton, however, thought that his main contribution to the world was the theory of quaternions, which was in fact largely ignored. Stewart clearly has a soft spot for quaternions and octonions, and later in the book describes how there is a revival of interest in these almost forgotten structures, which may yet play an important part in physics. The book goes on to the work of Sophus Lie, who applied the notion of a group to continuous systems, and to the work of Élie Cartan and Wilhelm Killing, who played an important part in the classification of groups. This is followed by a look at Einstein's theories of relativity and the symmetries of spacetime which they revealed. Of course Einstein is well known for his belief that the theories of physics had to be beautiful, and he spent much time searching for a unified theory which met his criteria. Stewart discusses the Kaluza-Klein theory, which attempted to unify the forces of nature by adding a fifth dimension to spacetime, but was left behind as interest in quantum theory grew. Stewart also looks at elementary particles and how they are classified using symmetry groups. The is followed by a chapter on superstring theory and the work of Edward Witten.
What the book doesn't do
I'll say here that I was expecting the book to take a somewhat different direction in the later chapters. I'd have liked more on the mathematical development of group theory following Galois, and how mathematicians develop a sense of what is beautiful. Aren't modular forms the epitome of symmetry? Then the link to physics then comes from (to quote Wigner) 'The Unreasonable Effectiveness of Mathematics in the Natural Sciences'. I'd would have also been nice to have a fuller explanation of gauge symmetry and of groups such as SU(2) and SO(3)- maybe that's because I've yet to find a satisfactory explanation of these topics in a popular work, and since Stewart does so well with Galois theory maybe he would succeed here too. I'm wary of criticising a book just because it doesn't include the topics I would like, but I did feel that the biographical information about the famous physicists of the early 20th century is well covered in other books, and that this one did start to wander a bit at the end. In my mind this would reduce it's chances of winning the 2008 Royal Society Prize for Science Books
I suppose the question about a book like this is how mathematical it should be, and I guess that if it had followed the path I described above then it would have had bit too much maths for many people's tastes. Going via the biographies of the mathematicians concerned allows the subject to be presented in a non-technical way and Stewart does well in making the book accessible to a wide readership. (Although it helps if the reader has a litle bit of mathematical knowledge)The book gives a wide ranging look at the history of some important parts of mathematics, and as long as you don't mind that it doesn't get on to some of the latest topics in mathematics then you're likely to find it to be a highly entertaining read.
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