New Books in Mathematics

The Story of Proof: Logic and the History of Mathematics (Princeton UP, 2022) investigates the evolution of the concept of proof--one of the most significant and defining features of mathematical thought--through critical episodes in its history. From the Pythagorean theorem to modern times, and across all major mathematical disciplines, John Stillwell demonstrates that proof is a mathematically vital concept, inspiring innovation and playing a critical role in generating knowledge.Stillwell begins with Euclid and his influence on the development of geometry and its methods of proof, followed by algebra, which began as a self-contained discipline but later came to rival geometry in its mathematical impact. In particular, the infinite processes of calculus were at first viewed as "infinitesimal algebra," and calculus became an arena for algebraic, computational proofs rather than axiomatic proofs in the style of Euclid. Stillwell proceeds to the areas of number theory, non-Euclidean geometry, topology, and logic, and peers into the deep chasm between natural number arithmetic and the real numbers. In its depths, Cantor, Gödel, Turing, and others found that the concept of proof is ultimately part of arithmetic. This startling fact imposes fundamental limits on what theorems can be proved and what problems can be solved.This book could well serve as a history of mathematics, because in developing the evolution of the concept of proof and how it has arisen in the various different mathematical fields. The author essentially traces the important milestones in the development of mathematics. It's an amazing job of collecting and categorizing many of the most important ideas in this area. Learn more about your ad choices. Visit megaphone.fm/adchoicesSupport our show by becoming a premium member! https://newbooksnetwork.supportingcast.fm/mathematics

We’ve all heard that a picture is worth a thousand words, but what if we don’t understand what we’re looking at? Social media has made charts, infographics, and diagrams ubiquitous―and easier to share than ever. We associate charts with science and reason; the flashy visuals are both appealing and persuasive. Pie charts, maps, bar and line graphs, and scatter plots (to name a few) can better inform us, revealing patterns and trends hidden behind the numbers we encounter in our lives. In short, good charts make us smarter―if we know how to read them.However, they can also lead us astray. Charts lie in a variety of ways―displaying incomplete or inaccurate data, suggesting misleading patterns, and concealing uncertainty―or are frequently misunderstood, such as the confusing cone of uncertainty maps shown on TV every hurricane season. To make matters worse, many of us are ill-equipped to interpret the visuals that politicians, journalists, advertisers, and even our employers present each day, enabling bad actors to easily manipulate them to promote their own agendas.In How Charts Lie: Getting Smarter about Visual Information (W. W. Norton, 2019), data visualization expert Alberto Cairo teaches us to not only spot the lies in deceptive visuals, but also to take advantage of good ones to understand complex stories. Public conversations are increasingly propelled by numbers, and to make sense of them we must be able to decode and use visual information. By examining contemporary examples ranging from election-result infographics to global GDP maps and box-office record charts, How Charts Lie demystifies an essential new literacy, one that will make us better equipped to navigate our data-driven world. Learn more about your ad choices. Visit megaphone.fm/adchoicesSupport our show by becoming a premium member! https://newbooksnetwork.supportingcast.fm/mathematics

In The New Era in American Mathematics, 1920-1950 (Princeton University Press, 2022) Karen Parshall explores the institutional, financial, social, and political forces that shaped and supported the American Mathematics community in the first half of the twentieth century. Drawing from extensive archival and primary-source research, Professor Parshall uncovers the key players in American mathematics who worked together to effect change. She highlights the educational, professional, philanthropic, and governmental entities that bolstered progress and uncovers the strategies implemented by American mathematicians in their quest for the advancement of knowledge. Through an examination of how the American Mathematical community asserted itself on the international state, The New Era in Mathematics, 1920-1950 shows the way one nation became the focal point for the field.Karen Hunger Parshall is the Commonwealth Professor of History and Mathematics at the University of Virginia. She is the author of James Joseph Sylvester: Jewish Mathematician in a Victorian World and the coauthor of Taming the Unknown: A History of Algebra from Antiquity to the Early Twentieth Century.Marc Goulet is Professor in mathematics and Associate Dean in the College of Arts and Sciences at the University of Wisconsin-Eau Claire. Learn more about your ad choices. Visit megaphone.fm/adchoicesSupport our show by becoming a premium member! https://newbooksnetwork.supportingcast.fm/mathematics

Peter Winkler has been collecting mathematical puzzles since childhood. He has had published two previous collections, and recently he compiled his largest curated collection to date. Mathematical Puzzles (A K Peters, 2021) also takes an alluring new approach to the genre: In the Roman-numbered front matter, 300+ puzzles are presented, roughly in order of increasing difficulty. Fuller discussions of the puzzles are then organized into 24 chapters according to the key insight that leads to their solutions. Each insight gets a brief mathematical treatment, and by the end of each chapter the reader is primed to appreciate an important and kindred mathematical result.In our interview, Dr. Winkler walked me through a handful of puzzles from two chapters. The first, "The Law of Small Numbers", gives a tongue-in-cheek name to the strategy of gaining insight from a smaller version of a larger problem. The puzzles lead up to a discussion of perfect matchings, whose small-number analogues reveal an elegant geometric solution. The second, "Infinite Choice", begins with some variations on hat-guessing puzzles, which turn out to admit surprisingly powerful strategies even among infinitudes of players! The chapter closes with an accessible peek at graph coloring problems.The book is delightfully designed with playful header fonts and illustrations by cartoonist Jess Johnson, through an equally delightful collaboration we took a moment to discuss. As Dr. Winkler aimed, i believe the book will appeal both to the puzzle enthusiast with a limited background in mathematics and to the mathematics enthusiast who, like myself, never really took to puzzles—as well, of course, to those dual enthusiasts whom the author epitomizes.Suggested companion works: How to Solve It: A New Aspect of Mathematical Method by Georg Pólya Martin Gardner’s Mathematical Games: The Entire Collection of his Scientific American Columns Peter Winkler is the William Morrill Professor of Mathematics and Computer Science at Dartmouth College and, for 2019–2020, the Distinguished Visiting Professor for the Public Dissemination of Mathematics at the National Museum of Mathematics. He is the author of 160 research papers, a dozen patents, two previous puzzle books, a book on cryptographic techniques in the game of bridge, and a portfolio of compositions for ragtime piano. Learn more about your ad choices. Visit megaphone.fm/adchoicesSupport our show by becoming a premium member! https://newbooksnetwork.supportingcast.fm/mathematics

In Community College Mathematics: Past, Present, and Future (CRC Press, 2022), Brian Cafarella addresses the key questions: How can we build a future model for community college gatekeeper math classes that is both successful and sustainable? Additionally, how can we learn from the past and the present to build such a model? From the 1970’s to the pandemic in the early 2020’s, the book uses interviews with 30 community college faculty members from seven community colleges to explore math curricula as well as trends, initiatives, teaching practices, and mandates that have impacted community college mathematics.Brian Cafarella is a professor in mathematics at Sinclair Community College in Dayton, Ohio. He has taught a variety of courses ranging from developmental math through pre-calculus. Brian is a past recipient of the Roeche Award for teaching excellence and a past recipient of the Ohio Magazine Award for excellence in education.Marc Goulet is Professor in mathematics and Associate Dean in the College of Arts and Sciences at the University of Wisconsin-Eau Claire. Learn more about your ad choices. Visit megaphone.fm/adchoicesSupport our show by becoming a premium member! https://newbooksnetwork.supportingcast.fm/mathematics

Today I had the pleasure of talking to Joe Mileti, associate professor of mathematics at Grinnell College. Even if you are not "into" math, you will enjoy this conversation. We talked about how math is not what you think it is. It's not just memorizing formulas and grinding. It's about thinking and, like all thinking, it involves abstraction, logic, using analogies and metaphors, and a bunch of imagination. We also talked about how math is about talking to other mathematicians and doing a kind of "brainstorming." Joes's new book is Modern Mathematical Logic (Cambridge UP, 2022).Marshall Poe is the founder and editor of the New Books Network. He can be reached at marshallpoe@newbooksnetwork.com. Learn more about your ad choices. Visit megaphone.fm/adchoicesSupport our show by becoming a premium member! https://newbooksnetwork.supportingcast.fm/mathematics

In Probability and Forensic Evidence: Theory, Philosophy, and Applications (Cambridge UP, 2021), Ronald Meester and Klaas Slooten address the role of statistics and probability in the evaluation of forensic evidence, including both theoretical issues and applications in legal contexts. It discusses what evidence is and how it can be quantified, how it should be understood, and how it is applied (and, sometimes misapplied).Ronald Meester is Professor in probability theory at the Vrije Universiteit Amsterdam. He is co-author of the books Continuum Percolation (1996), A Natural Introduction to Probability Theory (2003), Random Networks for Communication (2008), and has written around 120 research papers on topics including percolation theory, ergodic theory, philosophy of science, and forensic probability.Klaas Slooten works as Statistician at the Netherlands Forensic Institute and at the Vrije Universiteit Amsterdam where he is Professor by special appointment. He as published around 30 articles on forensic probability and statistics. He is interested in the mathematical, legal, and philosophical evaluation of evidence.Marc Goulet is Professor in mathematics and Associate Dean in the College of Arts and Sciences at the University of Wisconsin-Eau Claire. Learn more about your ad choices. Visit megaphone.fm/adchoicesSupport our show by becoming a premium member! https://newbooksnetwork.supportingcast.fm/mathematics

In Formulations: Architecture, Mathematics, Culture (MIT Press, 2022), Andrew Witt examines the visual, methodological, and cultural intersections between architecture and mathematics. The linkages Witt explores involve not the mystic transcendence of numbers invoked throughout architectural history, but rather architecture’s encounters with a range of calculational systems—techniques that architects inventively retooled for design. Witt offers a catalog of mid-twentieth-century practices of mathematical drawing and calculation in design that preceded and anticipated digitization as well as an account of the formal compendia that became a cultural currency shared between modern mathematicians and modern architects.Witt presents a series of extensively illustrated “biographies of method”—episodes that chart the myriad ways in which mathematics, particularly the mathematical notion of modeling and drawing, was spliced into the creative practice of design. These include early drawing machines that mechanized curvature; the incorporation of geometric maquettes—“theorems made flesh”—into the toolbox of design; the virtualization of buildings and landscapes through surveyed triangulation and photogrammetry; formal and functional topology; stereoscopic drawing; the economic implications of cubic matrices; and a strange synthesis of the technological, mineral, and biological: crystallographic design.Trained in both architecture and mathematics, Witt uses mathematics as a lens through which to understand the relationship between architecture and a much broader set of sciences and visual techniques. Through an intercultural exchange with other disciplines, he argues, architecture adapted not only the shapes and surfaces of mathematics but also its values and epistemic ideals.Bryan Toepfer, AIA, NCARB, CAPM is the Principal Architect for TOEPFER Architecture, PLLC, an Architecture firm specializing in Residential Architecture and Virtual Reality. He has authored two books, “Contractors CANNOT Build Your House,” and “Six Months Now, ARCHITECT for Life.” He is an Assistant Professor at Alfred State College and has served as the Director of Government Affairs and as the Director of Education for the AIA Rochester Board of Directors. Always eager to help anyone understand the world of Architecture, he hosts the New Books Network – Architecture podcast, is an NCARB Licensing Advisor and helps coach candidates taking the Architectural Registration Exam. btoepfer@toepferarchitecture Learn more about your ad choices. Visit megaphone.fm/adchoicesSupport our show by becoming a premium member! https://newbooksnetwork.supportingcast.fm/mathematics

Formal mathematical models have provided tremendous insights into politics in recent decades. Formal Models of Domestic Politics (Cambridge UP, 2021) is the leading graduate textbook covering the crucial models that underpin current theoretical and empirical research on politics by both economists and political scientists. This textbook was recently updated to reflect the wealth of new theory-building around the functioning of authoritarian regimes, as well as to include recent developments in the theory of electoral competition, delegation, legislative bargaining, and collective action.Author Scott Gehlbach is a professor at the University of Chicago, where he is the director of their new PhD program in Political Economy. He is a widely published political economist, with influential articles published in both economics and political science journal as well as two other books. Scott also has a regional specialty in the politics of Russia, Ukraine, and other postcommunist countries.In our interview, we discuss what formal models are, how they work, and illustrates their usefulness with several examples. We also speak briefly at the end about current Russian politics and the ideas he outlined in his February Washington Post Monkey Cage blog piece on the implications of the invasion of Ukraine.Host Peter Lorentzen is an Associate Professor in the Department of Economics at the University of San Francisco, where he leads a new Master's program in Applied Economics focused on the digital economy. His own research focus is the political economy of governance in China. Learn more about your ad choices. Visit megaphone.fm/adchoicesSupport our show by becoming a premium member! https://newbooksnetwork.supportingcast.fm/mathematics

Today I talked to Seth Stephens-Davidowitz about his new book Don't Trust Your Gut: Using Data to Get What You Really Want in LIfe (Dey Street Books, 2022)Looking for advice on how to get a date, how to have a successful marriage, or just how to have a happier life? Don’t trust your gut, don’t trust conventional wisdom, and put down that self-help book full of plausible arguments and compelling anecdotes that just happens to contradict the advice you got from the self-help book you. Instead, let Seth Stephens-Davidowitz guide you, using data!Seth Stephens-Davidowitz is a data scientist, author, keynote speaker, and recovering economist. His first book Everybody Lies, was a New York Times bestseller that showed how social scientists have used new data about our online behavior to gain new insights about who we really are and what we really think. His latest book, Don’t Trust Your Gut, is about how we can use data not just to understand other people but also how to get what we want in life, whether it’s health, wealth, attractiveness, or inner peace.Host Peter Lorentzen is an Associate Professor in the Department of Economics at the University of San Francisco, where he leads a new Master's program in Applied Economics focused on the digital economy. Learn more about your ad choices. Visit megaphone.fm/adchoicesSupport our show by becoming a premium member! https://newbooksnetwork.supportingcast.fm/mathematics