Spiegelhalter (Univ. of Cambridge) here offers a thoughtful introduction to statistics and data analysis with potential appeal for a wide audience. Topics addressed include distinguishing between effective and ineffective chart and graph implementations, with a thorough description of regression analysis, a deep dive into causation, an overview of artificial intelligence and prediction tools, and a more detailed look at causation and probability considered together. Examples of various statistical methods are interesting because they are based on situations familiar in daily life. Each chapter concludes with a bullet-point list of key concepts covered, a potential help especially for readers new to the processes and methods involved. The book’s glossary, index, and notes also provide helpful context for the target reader. The book is suitable for undergraduates and anyone seeking to improve their statistical literacy. The author’s aim is clearly to improve readers’ understanding of statistics and what they mean, without requiring them to actually perform statistical calculations. View on Amazon
Mathematical logic can be intimidating and usually is extraneous to the courses students earning math degrees take. To bridge this void, this text provides a whirlwind tour of the basics and highlights of mathematical logic. Building on the ideas of naïve set theory, the authors quickly expand to the richer areas of first-order logic, formal proofs, model theory, recursion theory, Gödel’s Incompleteness Theorems, and axiomatic set theory. Each chapter and every page shares important ideas—theorems, concepts, and proofs—related to an extremely wide variety of topics: ordinal vs. cardinal numbers, the axiom of choice, Zermelo-Fraenkel axiomatic systems, Goodstein sequences, infinite combinatorics, Henkin witnesses as a proof technique, Craig interpolation, Beth definability, quantifier elimination (allows proof of the Lefschetz principle in algebraic geometry), undecidability of the halting problem, Church and Tarski’s classical results on undecidability and incompleteness, a complete proof of Gödel’s second incompleteness theorem, and the von Neumann hierarchy. All of this in a 185-page book. View on Amazon
Although many books have treated the geometric questions raised by the early Greeks, this new work by Richeson (Dickinson College) offers a fresh and lively presentation of these same classical problems: doubling of the cube, angle trisection, squaring a circle, and construction of a regular heptagon. Happily, he offers a fuller, richer picture of their history, while also demonstrating their impossibility of solution by ruler and compass. Richeson’s text leads readers on a historical odyssey in quest of solutions to these and other problems, which in the end are shown to be unsolvable by classical methods. Among the many historical anecdotes included is the story of the 1655 dispute between Thomas Hobbes and John Wallis over Hobbes’s claim to have squared the circle—itself the subject of Jesseph’s earlier work (Squaring the Circle, CH, May’00, 37-5074). From ancient Greece to the dawn of modern algebra, Richeson charts the evolution of the mathematics related to these problems and explains the new mathematics needed to resolve them, in a masterful text covering not only the mathematical ideas but also their history. View on Amazon
As explained in the related Wikipedia article, the term stochastic denotes a “randomly determined process.” In this work, authors Erban and Chapman (both, Univ. of Oxford) provide an introduction to stochastic modeling that assumes basic familiarity with differential equations but not with probability theory or stochastic analysis per se. The text can be used effectively for solitary study or as a textbook for a course offered at the boundary between undergraduate and beginning graduate study. Throughout, the authors maintain balance and integration of stochastic and deterministic methods, guiding readers to master the computational techniques without assuming commitment to any particular programming language. Algorithms discussed are written in general pseudocode. A helpful supporting feature is an available online selection of MATLAB files to generate the copious figures essential for readers’ understanding (http://people.maths.ox.ac.uk/erban/cupbook/). Each chapter includes a section of exercises. Three helpful appendixes summarizing the basics of deterministic modeling and probability distributions accompany the rich set of references and useful index that conclude the book. This is a remarkable, even admirable, work that bears the mark of its Oxford origins. View on Amazon
At first glance, this slender volume might inspire comparison to P. Winkler’s Mathematical Puzzles (CH, Oct’04, 42-1000), but here, the puzzles are sewn into a fanciful Arthurian-style narrative, the better to appeal to young readers by avoiding the stench of schoolwork. However, now take the titular “unsolved” in its strongest sense: these 16 selected questions have indeed generally frustrated all attempts by any person (or machine) anywhere—and famously so. Though difficult-to-solve but simple-to-state challenges do occasionally yield, finally, to clever amateurs and child prodigies, bet with confidence that the chestnuts included here will long defeat all attempts (albeit some valuable mathematics might be created as fallout). The authors’ purpose seems actually to lie in communicating broadly the nature of pure mathematicians’ activity, something eluding even most graduating mathematics majors. View on Amazon
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