Why Beauty Is Truth: The History of Symmetry 1st Edition by Ian Stewart (PDF)

Description

At the heart of relativity theory, quantum mechanics, string theory, and much of modern cosmology lies one concept: symmetry. In Why Beauty Is Truth, world-famous mathematician Ian Stewart narrates the history of the emergence of this remarkable area of study. Stewart introduces us to such characters as the Renaissance Italian genius, rogue, scholar, and gambler Girolamo Cardano, who stole the modern method of solving cubic equations and published it in the first important book on algebra, and the young revolutionary Evariste Galois, who refashioned the whole of mathematics and founded the field of group theory only to die in a pointless duel over a woman before his work was published. Stewart also explores the strange numerology of real mathematics, in which particular numbers have unique and unpredictable properties related to symmetry. He shows how Wilhelm Killing discovered “Lie groups” with 14, 52, 78, 133, and 248 dimensions-groups whose very existence is a profound puzzle. Finally, Stewart describes the world beyond superstrings: the “octonionic” symmetries that may explain the very existence of the universe.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐What I liked about reading this book, is the way that Ian Stewart has interwoven the lives of the men and women of mathematics into this history of symmetry. It’s not just a bland description of the discoveries made over time, but an enlightening trek into the lives of the individuals that made these discoveries possible – their joys, heartaches, triumphs, and defeats. We really get to know the people behind the dreams.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.

⭐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.

⭐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 has many good points, and some drawbacks. I think my own lack of mathematical knowledge held me back from fully appreciating it. (I got A in O level maths in 1981. I enjoyed maths at school, and felt I was getting to the interesting bits when I was forced towards physics chemistry and biology for A levels- looking back I wish I had the chance to do all four subjects)The good points are that is well written with a clear narrative showing how our mathematical thinking has developed over time. It shows well how seemingly abstract problems lead on to many insights that may be interesting of themselves (pure maths) or may help solve practical problems. (applied maths) What seems like purely abstract mathematics may later turn out to be the route to new applied knowledge. The “unreasonable effectiveness” of mathematics is shown in many examples throughout the book. The discussion of the relationship between truth and beauty is well nuanced, and it seems likely that truth will be beautiful, and that a current “ugly” or “messy” formulation is one awaiting its simplification. At school I was just beginning to get the idea that graphs, coordinates, geometry, equations and matrices were all ways of expressing the same idea in different formats. This book shows how these relationships come about, and evolve out from one another.The drawbacks of the book for me was that the final 100 pages largely lost me. I got certain headline points, but I did not understand the ideas behind group theory, Lie groups, Hamilton’s work, Killing’s work. I think this is a reflection of my ignorance, not the author’s writing.My feeling about this book is that it would be a great read for someone studying maths at A level or university and wanting to get an idea of how maths has developed and where it is going. It would whet the appetite and encourage their studies.

⭐An interesting insight into the history of mathematics and mathematicians, but I was expecting to learn a bit more about the maths. I was hoping for an “idiot’s guide” to some of the topics covered in this book, but you really need to be a mathematican to understand some of the explanations in which case you don’t really need them. But still an enjoyable book.

⭐Accessible and written with great clarity and verve – this is one of the best introductions to symmetry for the non-specialist out there. Delivery was punctual and condition good.

⭐I started to read this book and realized that it is too advanced for me at the moment. I have retired now and am looking for stimulating reading. I realize that I have to start on much simpler concepts first. Will return to the book later on!

⭐I would recommend this book. Ian Stewart is great in explaining a very complex mathematical concepts behind symmetry and group theory. It is a very enjoyable and rewarding book to read and has sharpened my interest in this fascinating topic.

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