Discovering the nature of light: the science and the story

This is a rather original and unconventional book, truly different from what one could expect by simply considering the keywords appearing in the title. Remarkably creative and original, with a highly personal style from the author of the book, showing a capacity to find interesting connections following a clear thread stemming from set theory and relating numerous ideas and disciplines in physics and computing with diverse mathematical concepts. Amazingly, its narrative is excellent, allowing the reading of initially abstruse and tedious texts to become an enjoyable experience.A book title can be attractive simply for the keywords it contains. In this case, set theory, physics, and computingmake it tremendously intriguing to try to discoverwhat kind of connections its author has found between the different subjects. The diversity of topics that appear in the book is striking: an introduction to set theory, connections with stability and instability in classical mechanics, as well as chaos theory using set theory and the Cantor set, simple ideas of binary trees to illustrate ideas of hierarchical levels, fractals and dimensions, the laws of thermodynamics, entropy and information, different entropies, probabilities, cosmology, quantum mechanics, quantum statistical mechanics, general relativity, applications in cryptography and codes of error correction, applications in computational biology, quantum computing, Feynman diagrams, applications in C++ program structures, finite elementmethods, etc. Likewise, questions about whether the universe is infinite are raised, finding answers based on ideas from set theory and mathematical logic. The information covered in this book provides a broad and interdisciplinary overview of ideas derived from set theory, offering a rather systematic approach that can be useful for a better understanding of the underlying principles behind a large collection of ideas somehow scattered in the education of a physicist. As what concerns its contents, the book starts providing a basic introduction to set theory, and all kind of sets of numbers, with notions of cardinality and ordinals, finding insights from this perspective into several applications in physics and computer sciences. The author’s background in computing explains the interesting connections appearing with topics such as discrete mathematics, graph theory, binary trees, and combinatorics. He shows a deep background in physics, mathematics, and computer science, integrating different ideas in a coherent description according to a plan described previously in the contents. In addition, he has made a great effort as a sort of reflection aloud, in explaining with words in a rather informalway deep ideas ofmathematical physics. From this point of view, the number of mathematical formulas and graphs is not as excessive as it could be. The book is written in such a manner that all necessary information could be eventually found in the monograph, so that it can be considered selfcontained. The whole book is divided in eight parts, and it contains a total of 21 chapters, each one ending with a rich collection of simple exercises and questions,mostly attempting to help the reader to test their understanding. To get an idea, there are chapters with more than 100 questions. One relevant part uses the Cantor set and binary trees to introduce chaos theory from a geometrical point of view. The intensive use of binary trees to illustrate numerous concepts provide an original way of showing ideas of discrete mathematics and hierarchical structures, probably derived from the computer science style, that are used also to show applications in statistical physics, such as the Brownian motion, Newton’s binomial formulas, also providing a geometrical point of view in quantum mechanics, relativity, and statistical physics. Moreover, the idea of connecting set theory with graph theory leads to Feynman diagrams to illustrate quantum-mechanical processes. The part related to cryptography involves coding-decoding, the RSA key exchange, and a rich discussion on this area, as well as error correction codes with an application in computational biology. A full part is devoted to quantum computing and factoring algorithms. At the end, there is an excellent list of relevant references on nearly all the topics discussed in the monograph that can be of enormous use to the most discerning reader. A relevant question to consider is to discuss to which kind of public the book is aimed. Even though the book is not a textbook, the author suggests that the book could be used as such for undergraduate courses in mathematics, computer sciences or physics, depending on the choice of the different chapters. Certainly, the material could eventually be used by an instructor for a particular course, though in my opinion the book is of a different nature, and it would be more appropriate for a quiet and reflective reading connecting the different ideas exposed in the book. It belongs to this group of books that could be termed as thought-provoking for the capacity to establish common links among a diversity of concepts from different disciplines. From this viewpoint, the book could be attractive and of interest tomathematically oriented readers,mathematical physicists, theoretical physicists, theoretical computer scientists, applied mathematicians, as well as mathematicians interested in logic, set theory, and their applications.