Truth Proof

Constructive mathematics is based on the thesis that the meaning of a mathematical formula is given, not by its truth-conditions, but in terms of what constructions count as a proof of it. However, the meaning of the terms `construction' and `proof' has never been adequately explained (although Kriesel, Goodman and Martin-Löf have attempted axiomatisations). This monograph develops precise (though not wholly formal) definitions of construction and proof, and describes the algorithmic substructure underlying intuitionistic logic. Interpretations of Heyting arithmetic and constructive analysis are given. The philosophical basis of constructivism is explored thoroughly in Part I. The author seeks to answer objections from platonists and to reconcile his position with the central insights of Hilbert's formalism and logic. Audience: Philosophers of mathematics and logicians, both academic and graduate students, particularly those interested in Brouwer and Hilbert; theoretical computer scientists interested in the foundations of functional programming languages and program correctness calculi.

Few heavy metal acts survived the turmoil of the early 1990s music scene. Pantera was different. Instead of humoring the market, the band instead demanded that the audience come to them by releasing a series of fiercely uncompromising, platinum albums, including Vulgar Display of Power and Far Beyond Driven -- two #1 albums that, like Metallica's And Justice for All, sold millions of copies despite minimal airplay. Rex Brown's memoir is the definitive account of life inside one of rock's biggest bands, which succeeded against all odds but ultimately ended in tragedy when iconic lead guitarist Darrell "Dimebag" Abbott was murdered mid-performance by a deranged fan. This is a lucid account of the previously untold story behind one of the most influential bands in heavy metal history, written by the man best qualified to tell the truth about those incredible and often difficult years of fame and excess.

“If you want to build better health and a better future, this book makes an excellent tool kit.”—David A. Kessler, MD, author of The End of Overeating and former commissioner of the FDA It sometimes seems as if everyone around us is being diagnosed with a chronic illness—and that we might soon join them. In Disease-Proof, leading specialist in preventive medicine Dr. David Katz draws upon the latest scientific evidence and decades of clinical experience to explain how we can slash our risk of every major chronic disease—heart disease, cancer, stroke, diabetes, dementia, and obesity—by an astounding 80%. Dr. Katz arms us with skillpower: a proven, user-friendly set of tools that helps us make simple behavioral changes that have a tremendous effect on our health and well-being. Inspiring, groundbreaking, and prescriptive, Disease-Proof proves making lasting lifestyle changes is easier than we think.

According to the great mathematician Paul Erdös, God maintains perfect mathematical proofs in The Book. This book presents the authors candidates for such "perfect proofs," those which contain brilliant ideas, clever connections, and wonderful observations, bringing new insight and surprising perspectives to problems from number theory, geometry, analysis, combinatorics, and graph theory. As a result, this book will be fun reading for anyone with an interest in mathematics.

In case you are considering to adopt this book for courses with over 50 students, please contact [email protected] for more information. This introduction to mathematical logic starts with propositional calculus and first-order logic. Topics covered include syntax, semantics, soundness, completeness, independence, normal forms, vertical paths through negation normal formulas, compactness, Smullyan's Unifying Principle, natural deduction, cut-elimination, semantic tableaux, Skolemization, Herbrand's Theorem, unification, duality, interpolation, and definability. The last three chapters of the book provide an introduction to type theory (higher-order logic). It is shown how various mathematical concepts can be formalized in this very expressive formal language. This expressive notation facilitates proofs of the classical incompleteness and undecidability theorems which are very elegant and easy to understand. The discussion of semantics makes clear the important distinction between standard and nonstandard models which is so important in understanding puzzling phenomena such as the incompleteness theorems and Skolem's Paradox about countable models of set theory. Some of the numerous exercises require giving formal proofs. A computer program called ETPS which is available from the web facilitates doing and checking such exercises. Audience: This volume will be of interest to mathematicians, computer scientists, and philosophers in universities, as well as to computer scientists in industry who wish to use higher-order logic for hardware and software specification and verification.

Susan Haack brings her distinctive work in theory of knowledge and philosophy of science to bear on real-life legal issues.

This book is an introduction to the language and standard proof methods of mathematics. It is a bridge from the computational courses (such as calculus or differential equations) that students typically encounter in their first year of college to a more abstract outlook. It lays a foundation for more theoretical courses such as topology, analysis and abstract algebra. Although it may be more meaningful to the student who has had some calculus, there is really no prerequisite other than a measure of mathematical maturity.