Mathematics

10 reviews on the universal language.

Posted on March 25, 2021 in Hot Topic

book covers

The Best writing on mathematics, ed. by Mircea Pitici. Princeton, 2019. 272p bibl index ISBN 9780691198675, $85.00; ISBN 9780691198354 pbk, $24.95; ISBN 9780691197944 ebook, contact publisher for price.
Reviewed in CHOICE July 2020

This is the 10th volume of an acclaimed anthology series that was first published in 2010. While many potential readers might hold a belief that writings on mathematics must be obscure, boring, and difficult, the essays in this collection will rapidly dispel such belief as a myth. Pitici (Syracuse Univ.) has carefully selected thirteen articles which exemplify the best of mathematical exposition, addressing a diverse array of topics. Some of the covered topics are gerrymandering, 3-D optical illusions, mathematics for big data, unsolvable problems, the mechanization of mathematics, and the paradox named for the paradoxical mathematician Paul Erdős. The mathematical novice will find the material accessible, and professionals may get an introduction to areas outside their niche. This anthology provides a window into the beauty and diversity found in mathematical thought and the clarity with which it can be presented. Many readers will no doubt be motivated to examine previous volumes in this series. Pitici has a PhD in mathematics from Cornell University, and is working on a master’s degree in library science. Summing Up: Highly recommended. Lower-division undergraduates through faculty and professionals. General readers. —R. L. Pour, emeritus, Emory and Henry College

Chartier, Timothy. Math bytes: Google bombs, chocolate-covered pi, and other cool bits in computing. Princeton, 2014. 136p bibl index afp ISBN 9780691160603, $24.95.
Reviewed in CHOICE October 2014

A mathematical Pandora’s box released is perhaps the best way to describe this book. It overflows with ideas, flitting from one fascinating topic to the next, often without an apparent connection. The range is impressive. Chartier (Davidson College) covers topics such as using M&Ms and the sine curve to estimate a value of pi, using matrices to find celebrity images approximating one’s own face, determining equations that launch Angry Birds, using M&Ms in painting portraits, and using pictures of Beyoncé that create Sierpinski’s triangle. He also discusses the mathematics underlying Google PageRank and viral tweeting and polar transformations of Marilyn Monroe’s image. And this is but a small taste! Chartier spends enough time on a topic (“math byte”) to tantalize the reader. He uses his words and diagrams wisely, having honed both in presentations he has given in front of students and general audiences. The index is very selective (e.g., it does not even include “pi”). Everyone should find a “math byte” of interest in this book, and perhaps end up finding other topics of interest as well. Summing Up: Highly recommended. All readership levels. —J. Johnson, Western Washington University

Cheng, Eugenia. How to bake [pi]: an edible exploration of the mathematics of mathematics. Basic Books, 2015. 288p index ISBN 9780465051717, $27.50.
Reviewed in CHOICE December 2015

In this book, food recipes and cooking, both as a process and science, motivate a wide range of mathematical discussions. Overflowing with analogies and metaphors, the text forces a rethinking of understandings, knowledge, and beliefs within mathematics. The multiple views and reviews of simple mathematics demonstrate the need for caution in how one regards the doing and content of mathematics. Cheng (Univ. of Sheffield, UK) delves into a wide and diverse menu of mathematics, intentionally setting the stage for her argument that one can use category theory to “make difficult mathematics easy.” The text does raise interesting questions and perspectives regarding mathematicians’ use of abstraction, generalization, principles to support process, internal and external motivations, axiomatization, logic, proof, structure, and universal properties. At times, however, the author flits too fast from one idea to another or leaves a difficult mathematical idea unresolved, creating an aftereffect of half-baked notions that don’t satisfy the appetite. The text’s final chapters on category theory seem incomplete or oversimplified. Nonetheless, readers uninterested in the significant mathematical questions being raised can perhaps focus on the interesting recipes—provided in a mathematics book, no less! The UK paperback edition has the title Cakes, Custard and Category Theory: Easy Recipes for Understanding Complex Maths (Profile Books). Summing Up: Recommended. All readership levels. —J. Johnson, emeritus, Western Washington University

Devadoss, Satyan L. Mage Merlin’s unsolved mathematical mysteries, by Satyan Devadoss and Matt Harvey. MIT, 2020. 120p ISBN 9780262044080, $24.00.
Reviewed in CHOICE November 2020

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. Lower-level mathematics undergraduates should run through this book for improved intellectual orientation, specifically for debunking two common misconceptions: that mathematics has arrived at a mostly finished state, with working mathematicians merely excellent practitioners of known procedures, and that known mathematics solves all simple-sounding problems, leaving research-level questions necessarily far beyond the grasp of the merely curious. Summing Up: Highly recommended. All readers. —D. V. Feldman, University of New Hampshire

Eppstein, David. Forbidden configurations in discrete geometry. Cambridge, 2018. 230p bibl index ISBN 9781108423915, $105.00; ISBN 9781108439138 pbk, $39.99; ISBN 9781108542975 ebook, $32.00.
Reviewed in CHOICE November 2018

The study of finite subsets of the plane properly belongs to geometry when questions involve metric information—for example, magnitudes of distances and angles. Configurations arise by abstracting away metric details: number the points, and for any three points taken in order, it matters only whether they veer to the left, the right, or lie on a line. Configuration theory belongs to combinatorics: a given number of points support only a (difficult to enumerate!) finite number of configurations. Mathematics always studies the implications between properties objects may possess, but this introduction focuses on monotone properties: those a configuration never loses upon deletion of a point. Failure to possess certain fixed subconfigurations constitutes a monotone property, and conversely, one tries to understand monotone properties in terms of their minimal forbidden configurations. Well-known combinatorial subjects such as graph minors and permutation patterns follow parallel themes, but Eppstein (Univ. of California, Irvine) here offers the first systematic exposition of the present topic. The result is a first-class treatment: Eppstein deftly sells the subject to the uninitiated, yet carries it to depths experts will appreciate. A generous supply of diagrams gracefully projects many ideas, and the professional-quality design makes the reading experience a pleasure. Summing Up: Highly recommended. Lower-division undergraduates through faculty and professionals. —D. V. Feldman, University of New Hampshire

Garcia, Stephan Ramon. 100 years of math milestones: the Pi Mu Epsilon centennial collection, by Stephan Ramon Garcia and Steven J. Miller. American Mathematical Society, 2019. 581p bibl index ISBN 9781470436520 pbk, $60.00; ISBN 9781470453305 ebook, $60.00.
Reviewed in CHOICE December 2019

“Dedicated to the promotion of mathematics and recognition of students who successfully pursue mathematical understanding,” Pi Mu Epsilon (PME) regularly includes challenging problems in its journal to encourage engagement with members. For PME’s centennial in 2013, the journal’s problem editors created a special collection of 100 problems. Miller (Williams College), one of the editors, and Garcia (Pomona College) collected those problems, each one celebrating a year since 1913 with a discussion of a significant event in the development of mathematics, and revised and expanded them to include cross references where possible. The new collection is a tour de force of exposition and invention, allowing readers to delve deeply into some topics by weaving concepts from one entry with others. For example, the 1913 entry features Paul Erdös and his various problems on arithmetic progressions among the natural numbers. The 2004 entry returns to this idea with an account of the Green-Tao theorem, which settles Erdös’s conjecture about progressions among the primes. The authors make many more wonderful choices, producing a remarkable contribution to mathematical literature. It belongs in every library serving students of mathematics. Summing Up: Essential. Upper-division undergraduates through faculty; professionals. —J. McCleary, Vassar College

Ording, Philip. 99 variations on a proof. Princeton, 2019. 260p bibl index ISBN 9780691158839, $24.95; ISBN 9780691185422 ebook, contact publisher for price.
Reviewed in CHOICE August 2019

Virtually every mathematics student is required to take an introduction to mathematics or an introduction to mathematical proofs class. These courses vary widely in the topics covered, but all attempt to help students develop the ability to write a “good” mathematical proof. Students often find this intimidating, imagining that, when dealing with mathematics, there is only one correct answer. In this text Ording (Sarah Lawrence College) provides 99 different proofs of the same theorem. The theorem itself is a fairly simple statement about the roots of a particular cubic equation. However, in reading the numerous proofs given for this theorem, one begins to appreciate both the aesthetics of proofs and the number of acceptable approaches to writing them. The proofs presented range from ancient to the Renaissance to the modern, even including a physics-based proof using null points of an electric field. The reader will gain insight into mathematical literature as it has evolved over time and as it has been influenced by various philosophies and cultures. In particular, the book serves as an eloquent reminder that reading and writing mathematics is also a study in linguistics. Summing Up: Highly recommended. Undergraduates. —J. T. Zerger, Catawba College

Richeson, David S. Tales of impossibility: the 2000-year quest to solve the mathematical problems of antiquity. Princeton, 2019. 436p bibl index ISBN 9780691192963, $29.95; ISBN 9780691194233 ebook, contact publisher for price.
Reviewed in CHOICE May 2020

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. Novice mathematicians and professionals alike will enjoy this carefully written and entertaining book. Summing Up: Highly recommended. Lower-division undergraduates through faculty. Students enrolled in two-year technical programs. —R. L. Pour, emeritus, Emory and Henry College

Stinerock, Robert. Statistics with R: a beginner’s guide. SAGE Publishing, 2018. 369p index ISBN 9781473924895, $157.00; ISBN 9781473924901 pbk, $55.00.
Reviewed in CHOICE September 2018

The use of the R statistical programming package continues to spread quickly, and the number of good guides to using R seems to be keeping pace. This text deserves to be counted among the best of those books; it has a number of noteworthy features. Stinerock (New University of Lisbon, Portugal) begins each chapter with a list of learning objectives, offering the reader an overview of what will be discussed. Equal attention is paid to statistical principles and to the use of R for statistical tasks. A summary concludes each chapter, along with lists of definitions and R functions, which are useful for review. Plenty of worked examples add to the quality of the book. Stinerock also provides a variety of homework exercises, whose solutions are available at the book’s associated website. This site provides a wealth of supplemental material, including additional exercises with solutions and all datasets and R scripts from the book. This accessible guide deserves the highest consideration as a textbook for an introductory course in statistics and would be useful too for self-study. Summing Up: Essential. Lower-division undergraduates through faculty and professionals. —R. Bharath, emeritus, Northern Michigan University

Weissman, Martin H. An illustrated theory of numbers. American Mathematical Society, 2017. 323p bibl indexes ISBN 9781470434939, $69.00.
Reviewed in CHOICE February 2018

It is rare that a mathematics book can be described with this word, but Weissman’s An Illustrated Theory of Numbers is gorgeous! Weissman (Univ. of California, Santa Cruz) not only wrote a great textbook on number theory but also did so in a visually stunning way. The work is full of hundreds of beautiful visuals that complement the otherwise difficult subject matter. Any reader with a high school geometry and algebra background will be prepared to read, understand, and enjoy this text. Nonetheless, the text does get quite advanced, including topics beyond elementary number theory. Fortunately, Weissman prepares readers so they do not realize they have entered advanced mathematics. Although the book is not proof focused, it contains several proofs—all chosen specifically for their geometric approach and often deviating from more-standard proofs. Perhaps the work’s only weakness is that it is too geometric—meaning that the inclusion and omission of specific topics was decided solely upon which images could be included. Though math students might miss out on a few core principles offered in a traditional number theory course, most readers will love this work because they will be able to see numbers for the first time. Summing Up: Highly recommended. Lower-division undergraduates and above; general readers. —A. Misseldine, Southern Utah University

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