He explains the numerous ontological foundations that have been proposed for quantum field theory and lays out his dissatisfactions with each. Penrose has never been shy of controversy, and he launches right in, beginning with his reservations about the inflationary model of the Big Bang. Everything else has been prologue to this moment. The most interesting, in fact, delightful part of the book starts with chapter 28, page 735, when he begins his assessment of the state of modern physics. It enables a glimpse into the mind of Roger Penrose, how does he think about this mathematics and this physics? What are the concepts and understandings that he draws on as he uses these tools to attack the problems of modern physics? Before teaching any of these subjects, it would be worth reading the relevant chapter and reflecting on what Penrose sees as important, where he thinks the emphasis should be placed. The book is particularly effective for someone who already knows something or a lot about these subjects. You can't possibly learn these topics from this book, but Penrose does give very effective encapsulations of the key ideas. It is all here, from Minkowskian geometry to quantum field theory. Without pausing, he then begins using these tools to develop modern physics. In less than four hundred pages he has covered non-Euclidean geometries, single- and multi-variable as well as vector calculus, Fourier series, conformal mappings, the Cauchy-Riemann equations, Clifford, Grassman, and Lie algebras together with representation theory, the classical groups, calculus on manifolds, Clifford bundles and projective spaces, the Axiom of Choice, orders of infinity, Turing machines, and Gödel's Incompleteness Theorem. Penrose soon shifts to the solid ground of the development of mathematics and physics. Even Newton never succeeded in establishing the exact connection between the moon and tides. In addition, the effects of these forces are shaped by the geophysical structure of each basin. Cohen explains, tidal forces operate in three classes: diurnal (≈24 hours), semidiurnal (≈12 hours), and those with a period of half a month or more. In fact, this connection was not made until Newton's Principia of 1687, and it was one of the most controversial parts of the Principia. He uses as his illustrating example the connection between tides and phases of the moon. The one that really bothered me is on the first page of the first chapter in which Penrose sets the stage by explaining how science arose from observations of the regularities of the physical universe. It is the prerogative of a reviewer to focus on whatever nits he wishes to pick. The metaphysical speculation is covered in mercifully brief chapters at the beginning and end of the book. The fourth and most interesting part of the book is an extended discussion of the state of current attempts to find a grand unified theory. The third brings the reader up to an explanation of general relativity and quantum field theory. The second develops all of the mathematics needed to understand modern physics. The first consists of metaphysical speculation on the nature of truth and the relationship of mathematics to reality. There are four distinct parts to this book. Despite the presence of footnoted exercises, this is not a textbook. Calculus gets eighteen pages complex analysis twelve, though, to be fair, Riemann surfaces, conformal mappings, and the Riemann mapping theorem get a further sixteen. Eleven hundred pages isn't enough space in which to accomplish this. Its goal is to bring the reader up to the cutting edge of research in mathematical physics, to be able to understand the complexities of the search for the grand unified theory that will simultaneously explain general relativity and quantum mechanics. Even the existence of irrational numbers is treated in some detail. This book begins with an assumption of no more than a basic familiarity of arithmetic. But it will provide scholars for centuries to come with intriguing and often frustrating hints of what was known at the start of the 21 st century. Will this book reveal all of the physical laws of the universe that have been lost? No. Then, miraculously, one of them stumbles upon an intact copy of A Complete Guide to the Laws of the Universe. Wandering scholars search for shreds of documents from the distant past that might help them reconstruct this knowledge. Knowledge of mathematics, physics, and all sciences has been lost. Imagine a time in the distant future when humanity has slid into an extended dark age.
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