New Books in Mathematics

Plato’s Heaven: A User’s Guide is based on an in-depth filmed conversation between Howard Burton and James Robert Brown, Emeritus Professor of Philosophy at the University of Toronto. This wide-ranging conversation addresses a central theme in current philosophy: Platonism vs. Naturalism and provides accounts of both approaches to mathematics. The Platonist-Naturalist debate over mathematics is explored in a comprehensive fashion and also sheds light on non-mathematical aspects of a dispute that is central to contemporary philosophy. Thought experiments stand as a fascinating challenge to the necessity of data in the empirical sciences. Are these experiments, conducted uniquely in our imagination, simply rhetorical devices or communication tools or are they an essential part of scientific practice? This book also surveys the current state of this debate and explores new avenues of research into the epistemology of thought experiments.Howard Burton is the founder of the Ideas Roadshow, Ideas on Film and host of the Ideas Roadshow Podcast. He can be reached at howard@ideasroadshow.com. Learn more about your ad choices. Visit megaphone.fm/adchoicesSupport our show by becoming a premium member! https://newbooksnetwork.supportingcast.fm/mathematics

To many mathematicians and math enthusiasts, the word "innumeracy" brings to mind popular writing like that of John Allen Paulos. But inequities in our quantitative reasoning skills have received considerable interest and attention from researchers lately, including in psychology, development, education, and public health. Innumeracy in the Wild: Misunderstanding and Misusing Numbers (Oxford University Press, 2020) is a unified treatment of these broad-ranging studies, from the ways more and less numerate people differ in our perceptions of risk and our number-based decisions to the roots of our numeric faculties and how we can make the best of them. Dr. Ellen Peters has made significant contributions to the subject and brings her expertise and an exceptional clarity to its presentation.Precious little of the research surveyed in her book could fit into this interview! We discussed the three components of numeric ability—objective numeracy, subjective numeracy, and the innate number sense—and how they vary within and across populations. We talked through some key lessons from this literature, such as the importance of calibrating our self-efficacy to our real ability and an awareness of how our cultural allegiances can drive even our mathematical reasoning. And we identified some of the essential personal habits and policy levers (early childhood education!!) available to us in our efforts to improve our individual numeracy and our collective numeric decision-making. For a firm grounding in the state of knowledge and urgent open questions, there may be no better resource for many years to come.Suggested companion works: Contributions from the labs of Isaac Lipkus, Angela Fagerlin, John Opfer, Edward Cokely, Rocio Garcia-Retamero, Jakub Traczyk, Agata Sobków, Wändi Bruine de Bruin, Keith Stanovich, and Valerie Reyna.Ellen Peters, Ph.D., is the Philip H. Knight Chair, and Director of the Center for Science Communication Research, in the University of Oregon’s School of Journalism and Communication. As a decision psychologist, she studies the basic building blocks of human judgment and decision making and their links with effective communication techniques and has published more than 150 peer-reviewed papers on these topics. She is former President of the Society for Judgment and Decision Making and a fellow of the American Association for the Advancement of Science, Association for Psychological Science, the American Psychological Association, and the Society for Experimental Social Psychology. She also works with federal agencies to advance decision and communication sciences in health and health policy, including having been Chair of FDA’s Risk Communication Advisory Committee and member of the NAS’s Science of Science Communication committee. She has been awarded the Jane Beattie Scientific Recognition Award and an NIH Group Merit Award. Finally, she has received extensive funding from the National Science Foundation and National Institutes of Health. Cory Brunson is a Research Assistant Professor at the Laboratory for Systems Medicine at the University of Florida. His research focuses on geometric and topological approaches to the analysis of medical and healthcare data. He welcomes book suggestions, listener feedback, and transparent supply chains. Learn more about your ad choices. Visit megaphone.fm/adchoicesSupport our show by becoming a premium member! https://newbooksnetwork.supportingcast.fm/mathematics

Math circles defy simple narratives. The model was introduced a century ago, and is taking off in the present day thanks in part to its congruence with cutting-edge research in mathematics education. It is a modern approach to teaching—or facilitation—that resonates and finds mutual reinforcement with traditional practices and cultural preservation efforts. A wide range of math circle resources have become available for interested instructors, including the MSRI Math Circles Library, now in its 14th year of publication by the AMS.I was excited to talk with three editors and contributors to a recent volume in the series, Inspiring Mathematics: Lessons from the Navajo Nation Math Circles (American Mathematical, 2019). Drs. Dave Auckly, Amanda Serenevy, and Henry Fowler have been instrumental to the Navajo Nation Math Circles Project, along with co-editors Tatiana Shubin and Bob Klein and a broader contact and support network. Their book showcases scripts developed and facilitated in Navajo Nation, including an introduction to modular arithmetic through bean bag tossing, prefix sorting in the guise of pancake flipping, and a tactile use of limiting behavior to folding a necktie. We discussed the origin and expansion of math circles, their potential to indigenous mathematics educators and students, and the content of and stories behind a selection of the scripts.Dr. Fowler's foreword and the editors' introduction situate the math circles movement and the Navajo Nation Math Circles Project in history, geography, and culture. Each script begins with a (minimal!) list of the necessary materials and a student handout that invites explorations with them. A short survey of connections to deeper mathematics precedes each handout, and each is followed by an extensive teacher's guide with (illustrative) solutions and presentation suggestions. The scripts vary in complexity and are suitable for student- and teacher-focused math circles. I hope the text becomes widely adopted for science-based and culturally conscious mathematics education and helps introduce others like myself to the greater math circles project.Suggested companion works:-James Tanton-The Julia Robinson Mathematics Festival-Gordon Hamilton and Lora Saarnio, MathPickle-Robert Kaplan and Ellen Kaplan, Out of the Labyrinth: Setting Mathematics Free-Rachel and Rodi Steinig, Math RenaissanceDave Auckly is a research mathematician at Kansas State University and Co-founder and Director of the Navajo Nation Math Circles Project. Amanda Serenevy is Co-founder and Director of the Riverbend Community Math Center. Henry Fowler is Associate Professor of Mathematics at Navajo Technical University and Co-director of the Navajo Nation Math Circles Project. Cory Brunson is a Research Assistant Professor at the Laboratory for Systems Medicine at the University of Florida. Learn more about your ad choices. Visit megaphone.fm/adchoicesSupport our show by becoming a premium member! https://newbooksnetwork.supportingcast.fm/mathematics

Games have been of interest to mathematicians almost since mathematics became a subject. In fact, entire branches of mathematics have arisen simply to analyze certain games. The Raven's Hat: Fallen Pictures, Rising Sequences, and Other Mathematical Games (MIT Press, 2021) does something very different, and something that I think listeners will find intriguing – it uses games in order to explain mathematical concepts.The Raven's Hat presents a series of engaging games that seem unsolvable--but can be solved when they are translated into mathematical terms. How can players find their ID cards when the cards are distributed randomly among twenty boxes? By applying the theory of permutations. How can a player guess the color of her own hat when she can only see other players' hats? Hamming codes, which are used in communication technologies. Like magic, mathematics solves the apparently unsolvable. The games allow readers, including university students or anyone with high school-level math, to experience the joy of mathematical discovery. Learn more about your ad choices. Visit megaphone.fm/adchoicesSupport our show by becoming a premium member! https://newbooksnetwork.supportingcast.fm/mathematics

From its more mainstream, business-focused and business-friendly “Lean In” variants, to more radical, critical and intersectional understandings of feminism, the past decade has seen a flourishing of discussion from those proposing and critiquing different schools of thought for the way we think about gender in society.Dr. Eugenia Cheng’s addition to this conversation is x+y: A Mathematician's Manifesto for Rethinking Gender (Basic Books, 2020). She applies insights gained from her mathematical background to propose a new way to talk about gender and to propose an alternative: the terms “ingressive” and “congressive” behavior.In this interview, Dr. Cheng and I talk about what we gain from bringing a mathematical understanding to questions of social relations and structures. We talk about how she rethinks “gender”, and the new terms she proposes in her book. We end with a short discussion of whether these insights are applicable to conversations about other demographic and social identifiers.Dr. Eugenia Cheng is a mathematician and concert pianist. She is Scientist In Residence at the School of the Art Institute of Chicago and holds a PhD in pure mathematics from the University of Cambridge. Alongside her research in Category Theory and undergraduate teaching her aim is to rid the world of “math-phobia”. She was an early pioneer of math on YouTube and her videos have been viewed over 15 million times to date. Her other books are How to Bake Pi: An Edible Exploration of the Mathematics of Mathematics (Basic Books: 2016), which was featured on the Late Show with Stephen Colbert, Beyond Infinity: An Expedition to the Outer Limits of Mathematics (Basic Books: 2017) which was shortlisted for the Royal Society Science Book Prize in 2017 and The Art of Logic in an Illogical World (Basic Books: 2018)You can find more reviews, excerpts, interviews, and essays at The Asian Review of Books, including its review of x+y. Follow on Facebook or on Twitter at @BookReviewsAsia.Nicholas Gordon is a reviewer for the Asian Review of Books. In his day job, he’s a researcher and writer for a think tank in economic and sustainable development. He is also a print and broadcast commentator on local and regional politics. He can be found on Twitter at @nickrigordon. Learn more about your ad choices. Visit megaphone.fm/adchoicesSupport our show by becoming a premium member! https://newbooksnetwork.supportingcast.fm/mathematics

Jason Rosenhouse's Games for Your Mind: The History and Future of Logic Puzzles (Princeton UP, 2020) is about a panoply of logic puzzles. You’ll find Mastermind and sudoku discussed early on, and then you’ll be hit with an incredible array of some of the most intriguing logic puzzles that have ever been devised. Some will be familiar to you, but some will almost certainly be brain-teasers you have never heard of. It’s absolutely amazing what a truly deep field grew from recreational pastimes – and this book is an absolute treasure trove of stuff you can’t help thinking about. If you like logic, you’re certain to be sucked in – but you’ll enjoy the ride.Logic puzzles were first introduced to the public by Lewis Carroll in the late nineteenth century and have been popular ever since. Games like Sudoku and Mastermind are fun and engrossing recreational activities, but they also share deep foundations in mathematical logic and are worthy of serious intellectual inquiry. Games for Your Mind explores the history and future of logic puzzles while enabling you to test your skill against a variety of puzzles yourself.In this informative and entertaining book, Jason Rosenhouse begins by introducing readers to logic and logic puzzles and goes on to reveal the rich history of these puzzles. He shows how Carroll's puzzles presented Aristotelian logic as a game for children, yet also informed his scholarly work on logic. He reveals how another pioneer of logic puzzles, Raymond Smullyan, drew on classic puzzles about liars and truthtellers to illustrate Kurt Gödel's theorems and illuminate profound questions in mathematical logic. Rosenhouse then presents a new vision for the future of logic puzzles based on nonclassical logic, which is used today in computer science and automated reasoning to manipulate large and sometimes contradictory sets of data.Featuring a wealth of sample puzzles ranging from simple to extremely challenging, this lively and engaging book brings together many of the most ingenious puzzles ever devised, including the Hardest Logic Puzzle Ever, metapuzzles, paradoxes, and the logic puzzles in detective stories. Learn more about your ad choices. Visit megaphone.fm/adchoicesSupport our show by becoming a premium member! https://newbooksnetwork.supportingcast.fm/mathematics

It would be simple enough to say that mathematics is being done, and that those who do it are mathematicians. Yet, the history and culture of the mathematical community immediately complicate these statements. In her book A New Year's Present from a Mathematician (CRC Press, 2020), Snezana Lawrence guides a tour of European mathematical history that broadens conventional ideas of who mathematicians are and what we do. Framed as journey across the desert out from Alexandria, the book recounts a vignette from European mathematical history anchored to each month of the year, as drops of creativity and wisdom to sustain the trek.It is not unusual for books on the history of mathematics to tell very human stories about their often famous subjects. What is remarkable about Lawrence's collection is the breadth of these stories: Her subjects were idealists, pragmatists, mystics, skeptics, radicals, ascetics, and collectives. (Contrast, for example, the self-aware spite of Isaac Newton with the defiant good humor of Jean-Baptiste d'Alembert.) They contributed a mix each of original study, stewardship, and education. (Witness the precocious and persistent advocacy of Maria Agnesi and the devoted reciprocity of Johannes Kepler.) And they may or may not have been considered in their time, or even considered themselves, mathematicians.This book also showcases the diversity of mechanisms through which mathematics is transmitted and expanded. The projects undertaken by Lawrence's subjects are inspired by surviving ancient texts, popular treatments, and personal correspondence, and they yielded instructional texts, organizational schema, reference works, and popular fiction still in circulation today. The book drove home for me that the history of mathematics is ultimately a history of dialogue, and one that any person has the potential to contribute to—and thereby to be a mathematician.Suggested companion work: "Interstellar" (dir. Christopher Nolan)Snezana Lawrence is a mathematical historian, with a particular interest in the links between mathematics, architecture, and the belief systems related to mathematics. Her work on the creativity, identity, and engagement in the learning of mathematics has taken her to be involved in national and international initiatives to promote the use of the history of mathematics in mathematics education. She maintains the website Maths Is Good For You! and tweets @snezanalawrence. Cory Brunson is a Research Assistant Professor at the Laboratory for Systems Medicine at the University of Florida. Learn more about your ad choices. Visit megaphone.fm/adchoicesSupport our show by becoming a premium member! https://newbooksnetwork.supportingcast.fm/mathematics

Math has a complicated relationship with the counterintuitive: Rigorous logic, calculation, and simulation can both help us wrap our minds around phenomena that defy our intuition, and thrust upon us whole new worlds of counterintuitive results. In his new book, Jim Stein introduces readers to several unexpected and sometimes astonishing examples, while demanding a minimal mathematical background.The Fate of Schrodinger's Cat: Using Math and Computers to Explore the Counterintuitive (World Scientific, 2020) takes the reader along a journey in three segments. The first, through only by-hand calculations, builds up to a variation on Schrödinger's notorious thought experiment in which an observer can use an unrelated random process to predict the outcome of a 50/50 trial more than half the time. The kernel of this setup is Blackwell's Bet, a simple yet extraordinary illustration of what Stein calls "probabilistic entanglement".The second section uses computer simulation to get a handle on several paradoxical episodes in the world of sports: For my favorite example, how is it that an NFL season can at the same moment be exceptional both for the number of unbeaten teams and for the number of underperforming ones? Section III brings both computational approaches together to investigate perhaps the most argued-over quantitative question since Monty Hall: Is there a "hot hand"?What makes this book of popular mathematics exceptional is its openness: Stein's explorations can be followed with only very basic (or, ahem, BASIC) knowledge of arithmetic, probability, algebra, and programming. Moreover, they can be furthered: Readers are more often left not with final answers but with many ways to continue on their own.James D. Stein completed a BA in mathematics at Yale in 1962 and a PhD in mathematics at the University of California at Berkeley in 1967. He taught mathematics for 7 years at the University of California Los Angeles and for 35 years at California State University, Long Beach. His research has focused on Banach spaces and fixed-point theory, and he has written 10 mathematics and science books for the general public. He currently teaches one course per semester at El Camino Community College and is interested in probability theory and its applications to prediction.Cory Brunson is a Research Assistant Professor at the Laboratory for Systems Medicine at the University of Florida. His research focuses on geometric and topological approaches to the analysis of medical and healthcare data. He welcomes book suggestions, listener feedback, and transparent supply chains. Learn more about your ad choices. Visit megaphone.fm/adchoicesSupport our show by becoming a premium member! https://newbooksnetwork.supportingcast.fm/mathematics

Mathematics as a subject is distinctive in its symbolic abstraction and its potential for logical and computational rigor. But mathematicians tend to impute other qualities to our subject that set it apart, such as impartiality, universality, and elegance. Far from incidental, these ideas prime mathematicians and the public to see in mathematics the answers—for example, an impartial arbiter, or a meritocratic equalizer—to many urgent societal questions. Anna Weltman's new book, Supermath: The Power of Numbers for Good and Evil (Johns Hopkins University Press, 2020), surveys a number of ways this conception of mathematics has informed scientific undertakings and public policies, not to mention our everyday behaviors, and makes a powerful case for reevaluating its assumptions.The book's five chapters contain stories of mathematical exploits from ancient to ongoing and across the spectrum from pure to applied. Many may be familiar, for example active research and journalism into the use and misuse of predictive algorithms or G. H. Hardy's enumeration of the elements of mathematical beauty. Others, including continuing work to interpret Incan documents that survived European colonial erasure and the epidemiological insights obtained from massively multiplayer online gaming, will be new even to many mathematical readers. What they share is the essential but often ignored interplay between theory and culture that makes mathematics a thoroughly human activity. Weltman's book can be read as a call for scholars, educators, and communicators of mathematics to grapple with the power our training and credentialing in mathematics grants us, and to understand that its most basic promise of solving problems is not automatic but one that we must realize.Anna Weltman is a math teacher and writer who earned her PhD in mathematics education from the University of California at Berkeley. She is also the author of This Is Not a Math Book and This Is Not Another Math Book.Cory Brunson (he/him) is a Research Assistant Professor at the Laboratory for Systems Medicine at the University of Florida. Learn more about your ad choices. Visit megaphone.fm/adchoicesSupport our show by becoming a premium member! https://newbooksnetwork.supportingcast.fm/mathematics

Alfred S. Posamentier's The Joy of Geometry (Prometheus, 2020) is a book for someone who has taken geometry but wants to go further. This book, as one might expect, is heavy on diagrams and it is sometimes hard to discuss some of the ideas without reference to a diagram. Also, to be fair, this is not a book intended to be read casually. To fully appreciate this book, it is necessary to sit down, preferably in a comfortable chair with a beverage of one’s choosing, and prepare to give the diagrams a close look. The effort will be well rewarded. Learn more about your ad choices. Visit megaphone.fm/adchoicesSupport our show by becoming a premium member! https://newbooksnetwork.supportingcast.fm/mathematics