The Beauty of Mathematics in Computer Science explains the mathematical fundamentals of information technology products and services we use every day, from Google Web Search to GPS Navigation, and from speech recognition to CDMA mobile services. The book was published in Chinese in 2011 and has sold more than 600,000 copies. Readers were surprised to find that many daily-used IT technologies were so tightly tied to mathematical principles. For example, the automatic classification of news articles uses the cosine law taught in high school. The book covers many topics related to computer applications and applied mathematics including: Natural language processing Speech recognition and machine translation Statistical language modeling Quantitive measurement of information Graph theory and web crawler Pagerank for web search Matrix operation and document classification Mathematical background of big data Neural networks and Google's deep learning Jun Wu was a staff research scientist in Google who invented Google's Chinese, Japanese, and Korean Web Search Algorithms and was responsible for many Google machine learning projects. He wrote official blogs introducing Google technologies behind its products in very simple languages for Chinese Internet users from 2006-2010. The blogs had more than 2 million followers. Wu received PhD in computer science from Johns Hopkins University and has been working on speech recognition and natural language processing for more than 20 years. He was one of the earliest engineers of Google, managed many products of the company, and was awarded 19 US patents during his 10-year tenure there. Wu became a full-time VC investor and co-founded Amino Capital in Palo Alto in 2014 and is the author of eight books.

The language of ends and (co)ends provides a natural and general way of expressing many phenomena in category theory, in the abstract and in applications. Yet although category-theoretic methods are now widely used by mathematicians, since (co)ends lie just beyond a first course in category theory, they are typically only used by category theorists, for whom they are something of a secret weapon. This book is the first systematic treatment of the theory of (co)ends. Aimed at a wide audience, it presents the (co)end calculus as a powerful tool to clarify and simplify definitions and results in category theory and export them for use in diverse areas of mathematics and computer science. It is organised as an easy-to-cite reference manual, and will be of interest to category theorists and users of category theory alike.

The Beauty of Mathematics in Computer Science explains the mathematical fundamentals of information technology products and services we use every day, from Google Web Search to GPS Navigation, and from speech recognition to CDMA mobile services. The book was published in Chinese in 2011 and has sold more than 600,000 copies. Readers were surprised to find that many daily-used IT technologies were so tightly tied to mathematical principles. For example, the automatic classification of news articles uses the cosine law taught in high school. The book covers many topics related to computer applications and applied mathematics including: Natural language processing Speech recognition and machine translation Statistical language modeling Quantitive measurement of information Graph theory and web crawler Pagerank for web search Matrix operation and document classification Mathematical background of big data Neural networks and Google's deep learning Jun Wu was a staff research scientist in Google who invented Google's Chinese, Japanese, and Korean Web Search Algorithms and was responsible for many Google machine learning projects. He wrote official blogs introducing Google technologies behind its products in very simple languages for Chinese Internet users from 2006-2010. The blogs had more than 2 million followers. Wu received PhD in computer science from Johns Hopkins University and has been working on speech recognition and natural language processing for more than 20 years. He was one of the earliest engineers of Google, managed many products of the company, and was awarded 19 US patents during his 10-year tenure there. Wu became a full-time VC investor and co-founded Amino Capital in Palo Alto in 2014 and is the author of eight books.

This book presents reverse mathematics to a general mathematical audience for the first time. Reverse mathematics is a new field that answers some old questions. In the two thousand years that mathematicians have been deriving theorems from axioms, it has often been asked: which axioms are needed to prove a given theorem? Only in the last two hundred years have some of these questions been answered, and only in the last forty years has a systematic approach been developed. In Reverse Mathematics, John Stillwell gives a representative view of this field, emphasizing basic analysis--finding the "right axioms" to prove fundamental theorems--and giving a novel approach to logic. Stillwell introduces reverse mathematics historically, describing the two developments that made reverse mathematics possible, both involving the idea of arithmetization. The first was the nineteenth-century project of arithmetizing analysis, which aimed to define all concepts of analysis in terms of natural numbers and sets of natural numbers. The second was the twentieth-century arithmetization of logic and computation. Thus arithmetic in some sense underlies analysis, logic, and computation. Reverse mathematics exploits this insight by viewing analysis as arithmetic extended by axioms about the existence of infinite sets. Remarkably, only a small number of axioms are needed for reverse mathematics, and, for each basic theorem of analysis, Stillwell finds the "right axiom" to prove it. By using a minimum of mathematical logic in a well-motivated way, Reverse Mathematics will engage advanced undergraduates and all mathematicians interested in the foundations of mathematics.

Solow, "How to Read and Do Proofs, "provides a systematic approach for teaching students how to read, think about, understand, and create proofs. It develops a method for communicating proofs, categorizing, identifying, and explaining (at the student's level) the various techniques that are used repeatedly in virtually all proofs. These clear, concise explanations promote understanding of the theoretical mathematics behind abstract mathematics and give students a greater opportunity to succeed in advanced courses. Along with the addition of three new chapters, a "Part 2" is added to the Sixth Edition, which focuses on the mathematical thought processes associated with proofs. The teaching of this foregoing thinking processes reduces the time needed for readers to learn advanced mathematics courses while simultaneously increasing their depth of understanding so as to enable them to use mathematics more effectively as a problem-solving tool in their personal and professional lives.

In these essays Geoffrey Hellman presents a strong case for a healthy pluralism in mathematics and its logics, supporting peaceful coexistence despite what appear to be contradictions between different systems, and positing different frameworks serving different legitimate purposes. The essays refine and extend Hellman's modal-structuralist account of mathematics, developing a height-potentialist view of higher set theory which recognizes indefinite extendability of models and stages at which sets occur. In the first of three new essays written for this volume, Hellman shows how extendability can be deployed to derive the axiom of Infinity and that of Replacement, improving on earlier accounts; he also shows how extendability leads to attractive, novel resolutions of the set-theoretic paradoxes. Other essays explore advantages and limitations of restrictive systems - nominalist, predicativist, and constructivist. Also included are two essays, with Solomon Feferman, on predicative foundations of arithmetic.

What is the shortest possible route for a traveling salesman seeking to visit each city on a list exactly once and return to his city of origin? It sounds simple enough, yet the traveling salesman problem is one of the most intensely studied puzzles in applied mathematics--and it has defied solution to this day. In this book, William Cook takes readers on a mathematical excursion, picking up the salesman's trail in the 1800s when Irish mathematician W. R. Hamilton first defined the problem, and venturing to the furthest limits of today's state-of-the-art attempts to solve it. He also explores its many important applications, from genome sequencing and designing computer processors to arranging music and hunting for planets.

"In Pursuit of the Traveling Salesman" travels to the very threshold of our understanding about the nature of complexity, and challenges you yourself to discover the solution to this captivating mathematical problem.

The first book surveying the history and ideas behind reverse mathematics Reverse mathematics is a new field that seeks to find the axioms needed to prove given theorems. In Reverse Mathematics, John Stillwell offers a historical and representative view, emphasizing basic analysis and giving a novel approach to logic. By using a minimum of mathematical logic in a well-motivated way, Reverse Mathematics will engage advanced undergraduates and all mathematicians interested in the foundations of mathematics.

The last two decades have seen a wave of exciting new developments in the theory of algorithmic randomness and its applications to other areas of mathematics. This volume surveys much of the recent work that has not been included in published volumes until now. It contains a range of articles on algorithmic randomness and its interactions with closely related topics such as computability theory and computational complexity, as well as wider applications in areas of mathematics including analysis, probability, and ergodic theory. In addition to being an indispensable reference for researchers in algorithmic randomness, the unified view of the theory presented here makes this an excellent entry point for graduate students and other newcomers to the field.

In this entertaining and challenging collection of logic puzzles, Raymond Smullyan - author of Forever Undecided - continues to delight and astonish us with his gift for making available, in the thoroughly pleasurable form of puzzles, some of the most important mathematical thinking of our time. In the first part of the book, he transports us once again to that wonderful realm where knights, knaves, twin sisters, quadruplet brothers, gods, demons, and mortals either always tell the truth or always lie, and where truth-seekers are set a variety of fascinating problems. The section culminates in an enchanting and profound metapuzzle in which Inspector Craig of Scotland Yard gets involved in a search for the Fountain of Youth on the Island of Knights and Knaves. In the second part of To Mock a Mockingbird, we accompany the Inspector on a summer-long adventure into the field of combinatory logic (a branch of logic that plays an important role in computer science and artificial intelligence). His adventure, which includes enchanted forests, talking birds, bird sociologists, and a classic quest, provides for us along the way the pleasure of solving puzzles of increasing complexity until we reach the Master Forest and - thanks to Godel's famous theorem - the final revelation.

Reissuing works originally published between 1931 and 1990, this set of twenty-four books covers the full range of the philosophy of logic, from introductions to logic, to calculus and mathematical logic, to logic in language and linguistics and logical reasoning in law and ethics. An international array of authors are represented in this comprehensive collection.

Originally published in 1967. An introduction to the literature of nonstandard logic, in particular to those nonstandard logics known as many-valued logics. Part I expounds and discusses implicational calculi, modal logics and many-valued logics and their associated calculi. Part II considers the detailed development of various many-valued calculi, and some of the important metathereoms which have been proved for them. Applications of the calculi to problems in the philosophy are also surveyed. This work combines criticism with exposition to form a comprehensive but concise survey of the field.

Originally published in 1966. An introduction to current studies of kinds of inference in which validity cannot be determined by ordinary deductive models. In particular, inductive inference, predictive inference, statistical inference, and decision making are examined in some detail. The last chapter discusses the relationship of these forms of inference to philosophical notions of rationality. Special features of the monograph include a discussion of the legitimacy of various criteria for successful predictive inference, the development of an intuitive model which exhibits the difficulties of choosing probability measures over infinite sets, and a comparison of rival views on the foundations of probability in terms of the amount of information which the members of these schools believe suitable for fruitful formalization. The bibliographies include articles by statisticians accessible to students of symbolic logic.

Combining stories of great writers and philosophers with quotations and riddles, this text for first courses in mathematical logic examines problems related to proofs, propositional logic and first-order logic, undecidability, and other topics.

This book explores the results of applying empirical methods to the philosophy of logic and mathematics. Much of the work that has earned experimental philosophy a prominent place in twenty-first century philosophy is concerned with ethics or epistemology. But, as this book shows, empirical methods are just as much at home in logic and the philosophy of mathematics. Chapters demonstrate and discuss the applicability of a wide range of empirical methods including experiments, surveys, interviews, and data-mining. Distinct themes emerge that reflect recent developments in the field, such as issues concerning the logic of conditionals and the role played by visual elements in some mathematical proofs. Featuring leading figures from experimental philosophy and the fields of philosophy of logic and mathematics, this collection reveals that empirical work in these disciplines has been quietly thriving for some time and stresses the importance of collaboration between philosophers and researchers in mathematics education and mathematical cognition.

A classic exposition of a branch of mathematical logic that uses category theory, this text is suitable for advanced undergraduates and graduate students and accessible to both philosophically and mathematically oriented readers. Robert Goldblatt is Professor of Pure Mathematics at New Zealand's Victoria University. 1983 edition.

Already in just a decade of existence, cryptocurrencies have been the world's best-performing financial asset, outperforming stocks, bonds, commodities and currencies. This comprehensive yet concise book will enable the reader to learn about the nuts and bolts of cryptocurrencies, including their history, technology, regulations and economics. Additionally, this book teaches sound investment strategies that already work along with the spectrum of risks and returns. This book provides a plain-language primer for beginners worldwide on how to confidently navigate the rapidly evolving world of cryptocurrencies. Beginning by cutting to the chase, the author lists the common burning questions about cryptocurrency and provides succinct answers. Next, he gives an overview of cryptocurrency's underlying technology: blockchain. He then explores the history of cryptocurrency and why it's attracted so much attention. With that foundation, readers will be ready to understand how to invest in cryptocurrency: how cryptocurrency differs from traditional investments such as stocks, how to decide which cryptocurrency to invest in, how to acquire it, how to send and receive it, along with investment strategies. Additionally, legal issues, social implications, cybersecurity risks and the vocabulary of cryptocurrency are also covered, including Bitcoin and the many alternative cryptocurrencies. Written by a journalist-turned-professor, this book's appeal lies in its succinct, informative and easy-to-understand style. It will be of great interest to anyone looking to further their understanding of what cryptocurrency is, why it's a big deal, how to acquire it, how to send and receive it, and investment strategies.