Have you ever wondered what a mathematical language of trees would be like? Would you like to know how it is linked to homeomorphically irreducible trees, and the case n = 10? Perhaps you are more interested in some experimental theorems, like the theorem of sum forms? Or some famous numbers, like Euler's number and pi? How about some fun experimentation with the Collatz conjecture? Or maybe you are an odd one and want to see some possibly controversial, but not entirely pseudo-mathematical ideas? Should your answer to any of these questions be yes, then join this experimentally wonderful journey to mathematical truth!
A brilliant tour of mathematical thought and a guide to becoming a better thinker, How Not to Be Wrong shows that math is not just a long list of rules to be learned and carried out by rote. Math touches everything we do; It's what makes the world make sense. Using the mathematician's methods and hard-won insights-minus the jargon-professor and popular columnist Jordan Ellenberg guides general readers through his ideas with rigor and lively irreverence, infusing everything from election results to baseball to the existence of God and the psychology of slime molds with a heightened sense of clarity and wonder. Armed with the tools of mathematics, we can see the hidden structures beneath the messy and chaotic surface of our daily lives. How Not to Be Wrong shows us how--Publisher's description.
Part I of this book is a practical introduction to working with the Isabelle proof assistant. It teaches you how to write functional programs and inductive definitions and how to prove properties about them in Isabelle’s structured proof language. Part II is an introduction to the semantics of imperative languages with an emphasis on applications like compilers and program analysers. The distinguishing feature is that all the mathematics has been formalised in Isabelle and much of it is executable. Part I focusses on the details of proofs in Isabelle; Part II can be read even without familiarity with Isabelle’s proof language, all proofs are described in detail but informally. The book teaches the reader the art of precise logical reasoning and the practical use of a proof assistant as a surgical tool for formal proofs about computer science artefacts. In this sense it represents a formal approach to computer science, not just semantics. The Isabelle formalisation, including the proofs and accompanying slides, are freely available online, and the book is suitable for graduate students, advanced undergraduate students, and researchers in theoretical computer science and logic.
'Science has never had an advocate quite like David Deutsch ... A computational physicist on a par with his touchstones Alan Turing and Richard Feynman, and a philosopher in the line of his greatest hero, Karl Popper. His arguments are so clear that to read him is to experience the thrill of the highest level of discourse available on this planet and to understand it' Peter Forbes, Independent In our search for truth, how far have we advanced? This uniquely human quest for good explanations has driven amazing improvements in everything from scientific understanding and technology to politics, moral values and human welfare. But will progress end, either in catastrophe or completion - or will it continue infinitely? In this profound and seminal book, David Deutsch explores the furthest reaches of our current understanding, taking in the Infinity Hotel, supernovae and the nature of optimism, to instill in all of us a wonder at what we have achieved - and the fact that this is only the beginning of humanity's infinite possibility. 'This is Deutsch at his most ambitious, seeking to understand the implications of our scientific explanations of the world ... I enthusiastically recommend this rich, wide-ranging and elegantly written exposition of the unique insights of one of our most original intellectuals' Michael Berry, Times Higher Education Supplement 'Bold ... profound ... provocative and persuasive' Economist 'David Deutsch may well go down in history as one of the great scientists of our age' Scotsman
Typically, undergraduates see real analysis as one of the most difficult courses that a mathematics major is required to take. The main reason for this perception is twofold: Students must comprehend new abstract concepts and learn to deal with these concepts on a level of rigor and proof not previously encountered. A key challenge for an instructor of real analysis is to find a way to bridge the gap between a student’s preparation and the mathematical skills that are required to be successful in such a course. Real Analysis: With Proof Strategies provides a resolution to the "bridging-the-gap problem." The book not only presents the fundamental theorems of real analysis, but also shows the reader how to compose and produce the proofs of these theorems. The detail, rigor, and proof strategies offered in this textbook will be appreciated by all readers. Features Explicitly shows the reader how to produce and compose the proofs of the basic theorems in real analysis Suitable for junior or senior undergraduates majoring in mathematics.
The volume contains a comprehensive and problem-oriented presentation of ancient Greek mathematics from Thales to Proklos Diadochos. Exemplarily, a cross-section of Greek mathematics is offered, whereby also such works of scientists are appreciated in detail, of which no German translation is available. Numerous illustrations and the inclusion of the cultural, political and literary environment provide a great spectrum of the history of mathematical science and a real treasure trove for those seeking biographical and contemporary background knowledge or suggestions for lessons or lectures. The presentation is up-to-date and realizes tendencies of recent historiography. In the new edition, the central chapters on Plato, Aristotle and Alexandria have been updated. The explanations of Greek calculus, mathematical geography and mathematics of the early Middle Ages have been expanded and show new points of view. A completely new addition is a unique illustrated account of Roman mathematics. Also newly included are several color illustrations that successfully illustrate the book's subject matter. With more than 280 images, this volume represents a richly illustrated history book on ancient mathematics.
Introduction to Formal Languages, Automata Theory and Computation presents the theoretical concepts in a concise and clear manner, with an in-depth coverage of formal grammar and basic automata types. The book also examines the underlying theory and principles of computation and is highly suitable to the undergraduate courses in computer science and information technology. An overview of the recent trends in the field and applications are introduced at the appropriate places to stimulate the interest of active learners.
