Proven Impossible: Elementary Proofs of Profound Impossibility from Arrow, Bell, Chaitin, Gödel, Turing and More

In mathematics, it simply is not true that 'you can't prove a negative'. Many revolutionary impossibility theorems reveal profound properties of logic, computation, fairness and the universe, and form the mathematical background of new technologies and Nobel prizes. But to fully appreciate these theorems and their impact on mathematics and beyond, you must understand their proofs. This book is the first to present these proofs for a broad, lay audience. It fully develops the simplest rigorous proofs found in the literature, reworked to contain less jargon and notation, and more background, intuition, examples, explanations, and exercises. Amazingly, all of the proofs in this book involve only arithmetic and basic logic – and are elementary, starting only from first principles and definitions. Very little background knowledge is required, and no specialized mathematical training – all you need is the discipline to follow logical arguments and a pen in your hand.

Table of Contents:
Preface; 1. Yes you can prove a negative!; 2. Bell's impossibility theorem(s); 3. Enjoying Bell magic; 4. Arrow's (and friends') impossibility theorems; 5. Clustering and impossibility; 6. Gödel-ish impossibility; 7. Turing undecidability and incompleteness; 8. Chaitin's theorem: More devastating; 9. Gödel (for real, this time).

About the Author :
Dan Gusfield is Distinguished Professor emeritus, and former department chair, in the Computer Science Department at University of California, Davis. He is a Fellow of the ACM, the IEEE, and the ISCB. His previous books are 'The Stable Marriage Problem' (1989, co-authored with Rob Irving); 'Strings, Trees and Sequences' (1997); 'ReCombinatorics' (2014); and 'Integer Linear Programming in Computational and Systems Biology' (2019). As this book reflects, his teaching emphasized mathematical rigor as well as accessibility and clarity. He produced over 100 hours of video lectures on a wide range of topics, now viewed over a million times on the web.

Review :
'This unique and lovely book takes us on a grand tour of the limitations of science, mathematics, and of reason itself. To appreciate what is possible we must know the impossible, and such limitations define the boundary between the two. Gusfield offers well-explained gems illustrating various limitations, showing why they arise, giving their historical context, and in contrast to other similar books for a broad audience, presenting rigorous proofs requiring limited background.' Michael Sipser, MIT 'There are impossible problems in many different fields (e.g., Physics, Mathematics). This book is an excellent exposition of these difference ways a problem can be impossible. Along the way, the reader will pick up the needed background which is interesting in itself.' William Gasarch, University of Maryland 'This mathematics text is not the norm. It has an intriguing title, interesting content, and an author who expertly guides readers through difficult material. … Highly recommended.' J. Johnson, CHOICE '… a great book both for the layperson and for people who know some of the material.' William Gasarch, SIGACT News 'It is always a pleasure to stumble across a mathematics book that seems to be genuinely unique, one that offers the reader something not easily found elsewhere … most readers will not only find the book accessible but fun to read. The author takes pains to discuss topics slowly and in a well motivated way. The book also benefits from an extensive bibliography and end-of-chapter exercises, which struck me as well-chosen and helpful. The exercises enhance the value of this book as a text; I think a 'special topics' seminar course based on it, for example, would be fun for both students and faculty … Overall, this is an excellent book, one that is highly recommended … The author enjoyed writing it, and readers will enjoy reading it.' Mark Hunacek, The Mathematical Gazette