New Books in Mathematics

David S. Richeson's book Tales of Impossibility: The 2000-Year Quest to Solve the Mathematical Problems of Antiquity (Princeton University Press, 2019) is the fascinating story of the 2000 year quest to solve four of the most perplexing problems of antiquity: squaring the circle, duplicating the cube, trisecting the angle, and constructing regular polygons. The eventual conclusion was that all four of these problems could not be solved under the conditions laid out millennia ago. But it's also an engaging tale of some of the greatest mathematicians, and some not-so-well known ones, who met the challenge and moved mathematics forward in ways that the Greek geometers could never have envisioned. Even if you never read a single proof through to its conclusion, you'll enjoy the many entertaining side trips into a geometry far beyond what you learned in high school. Learn more about your ad choices. Visit megaphone.fm/adchoicesSupport our show by becoming a premium member! https://newbooksnetwork.supportingcast.fm/mathematics

Graduate students in many programs besides mathematics will need to be familiar with the methods and results of a variety of mathematical topics. Just as importantly, these students will need to develop a level of mathematical maturity—how to think in broad strokes about the subject, how to study it, and even how to communicate their work. The gap between undergraduate training and graduate requirements can also give rise to misconceptions about how mathematics is done and what needs drive cutting-edge research.Thomas A. Garrity's book All the Math You Missed (But Need to Know for Graduate School), now in its second edition (Cambridge UP, 2021), is a wide bridge across this gap—i do believe the material will bring any rising graduate student to readiness in the subject, whatever their needs. But it provides much more: Readers gain an intuition about the results and how they cohere that may be years coming through coursework alone; and an exposure to the culture of the mathematics community through historical vignettes, folklore, and prevailing beliefs (e.g. on the Riemann Hypothesis and whether P = NP). For such a broad introduction, the book is also surprisingly self-contained, and the chapters chart a uniformly continuous path through its topics.In our interview, Dr. Garrity recounted the origins of the book, described his outlook on undergraduate and graduate mathematics, and elaborated on the style(s) he adopted across its topics. We also got into the choices of content and depth he made and what role both editions have played over 23 years. The discussion resonated with several of my own experiences, and in any event i plan to keep my copy of the book handy for any (undergraduate or graduate) students i teach or mentor in future.Suggested companion works:• Saunders MacLane, Mathematics Form and Function• Timothy Gowers, June Barrow-Green, and Imre Leader (ed.), The Princeton Companion to MathematicsThomas A. Garrity is the Webster Atwell Class of 1921 Professor of Mathematics at Williams College, Massachusetts, where he was the director of the Williams College Project for Effective Teaching for many years. Among his awards are Rice University's Nicolas Salgo Outstanding Teaching award and the Haimo award of the MAA. His other books include Algebraic Geometry: A Problem Solving Approach (2013, co-authored) and Electricity and Magnetism for Mathematicians (2015). Learn more about your ad choices. Visit megaphone.fm/adchoicesSupport our show by becoming a premium member! https://newbooksnetwork.supportingcast.fm/mathematics

The stereotype of the solitary mathematician is widespread, but practicing users and producers of mathematics know well that our work depends heavily on our historical and contemporary fellow travelers. Yet we may not appreciate how our work also extends beyond us into our physical and societal environments.Kevin Lambert takes what might be a first crack at this perspective in his book Symbols and Things: Material Mathematics in the Eighteenth and Nineteenth Centuries (University of Pittsburgh Press, 2021). An historian of science, Dr. Lambert has shifted in his view of mathematics as a language of science to one as a material practice. Expanding on ideas from historians, archeologists, philosophers, and other scholars of human activity, and through several interweaving vignettes of mathematical work during a technologically dynamic period in British history, he argues that mathematical practice, communication, and even thought occur to a large degree outside the bodies of the persons performing them.In this interview, we explore Kevin's journal to and through this book project. We discuss how such ideas as Andy Clark's extended mind informed his approach, and we review several of the lively stories—the co-creation of the long-distance mathematical community with the research journal, Peacock's museological argument for the adoption of symbolic algebra, and the foundational entanglement of electromagnetism, quaternions, and the philosophy of space, among others—he drew out of historical and archival sources. (Here i cannot resist mentioning Tait's collection of his intensive correspondence with Hamilton that transformed how quaternions were applied in physics and even conceptualized as mathematical objects.) We close with some thoughts on our own materially extended cognitive work and where Kevin's interests are currently driving him.Suggested companion works:• ChatGPT, as a cutting-edge extension of human thought• work by Courtney Ann Roby, including the forthcoming The Mechanical Tradition of Hero of Alexandria: Strategies of Reading from Antiquity to the Early Modern Period• Algorithmic Modernity: Mechanizing Thought and Action, 1500-2000, edited by Morgan G. Ames and Massimo Mazzotti• work by Emily Miller Bonney, for example "A Reconsideration of Depositional Practices in Early Bronze Age Crete"Kevin Lambert is a historian of science and mathematics in the early modern and modern periods and professor in the liberal studies department at California State University, Fullerton. His recent book Symbols and Things explores mathematics as a way of thinking outside the body and through the material environment. He also recently published a chapter in the volume Algorithmic Modernity that traces the genealogy of algorithmic practices. He is now working on the problem of writing longue durée histories of science. He is close to completing a paper called “Malthus in the Landscape” that investigates the temporalities of global histories. He is also exploring the problem of writing a global history of the early modern sciences without the prism of the so called “Scientific Revolution.” His work can be found on ResearchGate. Learn more about your ad choices. Visit megaphone.fm/adchoicesSupport our show by becoming a premium member! https://newbooksnetwork.supportingcast.fm/mathematics

What's the best way to determine what most voters want when multiple candidates are running? What's the fairest way to allocate legislative seats to different constituencies? What's the least distorted way to draw voting districts? Not the way we do things now. Democracy is mathematical to its very foundations. Yet most of the methods in use are a historical grab bag of the shortsighted, the cynical, the innumerate, and the outright discriminatory. Making Democracy Count: How Mathematics Improves Voting, Electoral Maps, and Representation (Princeton UP, 2024) sheds new light on our electoral systems, revealing how a deeper understanding of their mathematics is the key to creating civic infrastructure that works for everyone.In this timely guide, Ismar Volic empowers us to use mathematical thinking as an objective, nonpartisan framework that rises above the noise and rancor of today's divided public square. Examining our representative democracy using powerful clarifying concepts, Volic shows why our current voting system stifles political diversity, why the size of the House of Representatives contributes to its paralysis, why gerrymandering is a sinister instrument that entrenches partisanship and disenfranchisement, why the Electoral College must be rethought, and what can work better and why. Volic also discusses the legal and constitutional practicalities involved and proposes a road map for repairing the mathematical structures that undergird representative government.Making Democracy Count gives us the concrete knowledge and the confidence to advocate for a more just, equitable, and inclusive democracy. Learn more about your ad choices. Visit megaphone.fm/adchoicesSupport our show by becoming a premium member! https://newbooksnetwork.supportingcast.fm/mathematics

Andy Hunter, founder and CEO of Bookshop.org, discusses his journey to creating the revolutionary organization that donates over 80% of its profits to independent bookstores. Learn about the challenges faced and how to support local bookstores through Bookshop.org. They also talk about the value of bookstores, personal recommendations, and their love for math and novels.

Barbara M. Sattler's book The Concept of Motion in Ancient Greek Thought: Foundations in Logic, Method, and Mathematics (Cambridge UP, 2020) examines the birth of the scientific understanding of motion. It investigates which logical tools and methodological principles had to be in place to give a consistent account of motion, and which mathematical notions were introduced to gain control over conceptual problems of motion. It shows how the idea of motion raised two fundamental problems in the 5th and 4th century BCE: bringing together being and non-being, and bringing together time and space. The first problem leads to the exclusion of motion from the realm of rational investigation in Parmenides, the second to Zeno's paradoxes of motion. Methodological and logical developments reacting to these puzzles are shown to be present implicitly in the atomists, and explicitly in Plato who also employs mathematical structures to make motion intelligible. With Aristotle we finally see the first outline of the fundamental framework with which we conceptualise motion today.Professor Barbara Sattler is Chair in Ancient and Medieval Philosophy at the Ruhr University Bochum. The main area of her research is metaphysics and natural philosophy in the ancient Greek world.Morteza Hajizadeh is a Ph.D. graduate in English from the University of Auckland in New Zealand. His research interests are Cultural Studies; Critical Theory; Environmental History; Medieval (Intellectual) History; Gothic Studies; 18th and 19th Century British Literature. YouTube channel. Learn more about your ad choices. Visit megaphone.fm/adchoicesSupport our show by becoming a premium member! https://newbooksnetwork.supportingcast.fm/mathematics

A defining feature of nineteenth-century Britain was its fascination with statistics. The processes that made Victorian society, including the growth of population, the development of industry and commerce, and the increasing competence of the state, generated profuse numerical data. Victorians and Numbers: Statistics and Society in Nineteenth Century Britain (Oxford UP, 2022) is a study of how such data influenced every aspect of Victorian culture and thought, from the methods of natural science and the struggle against disease, to the development of social administration and the arguments and conflicts between social classes. Numbers were collected in the 1830s by newly-created statistical societies in response to this 'data revolution'. They became a regular aspect of governmental procedure thereafter, and inspired new ways of interrogating both the natural and social worlds. William Farr used them to study cholera; Florence Nightingale deployed them in campaigns for sanitary improvement; Charles Babbage was inspired to design and build his famous calculating engines to process them. The mid-Victorians employed statistics consistently to make the case for liberal reform. In later decades, however, the emergence of the academic discipline of mathematical statistics - statistics as we use them today - became associated with eugenics and a contrary social philosophy. Where earlier statisticians emphasised the unity of mankind, some later practitioners, following Francis Galton, studied variation and difference within and between groups. In chapters on learned societies, government departments, international statistical collaborations, and different Victorian statisticians, Victorians and Numbers traces the impact of numbers on the era and the intriguing relationship of Victorian statistics with 'Big Data' in our own age.Lawrence Goldman was born in London and educated at Cambridge and Yale. Following a Junior Research Fellowship at Trinity College, Cambridge, he taught British and American History for three decades in Oxford, where he was a fellow of St. Peter's College, and Editor of the Oxford Dictionary of National Biography 2004-2014. Latterly he was Director of the Institute of Historical Research, University of London. His publications include books on Victorian social science and the history of workers' education, and a biography of the historian and political thinker R. H. Tawney. He is now Emeritus Fellow of St. Peter's College, Oxford.Morteza Hajizadeh is a Ph.D. graduate in English from the University of Auckland in New Zealand. His research interests are Cultural Studies; Critical Theory; Environmental History; Medieval (Intellectual) History; Gothic Studies; 18th and 19th Century British Literature. YouTube channel. Learn more about your ad choices. Visit megaphone.fm/adchoicesSupport our show by becoming a premium member! https://newbooksnetwork.supportingcast.fm/mathematics

One of the questions I am often asked is exactly what do mathematicians do. The short answer is that they look at different mathematical structures, try to deduce their properties, and think about how they might apply to the real world. Math Without Numbers (Dutton, 2020) does a wonderful job of explaining what mathematical structures are, and does so in a fashion that even readers who are uncomfortable with the process of doing mathematics can appreciate and enjoy. There are courses in music and art appreciation, and if there ever are courses in math appreciation, this book would certainly be at or near the top of the reading list. Learn more about your ad choices. Visit megaphone.fm/adchoicesSupport our show by becoming a premium member! https://newbooksnetwork.supportingcast.fm/mathematics

Many books explain what is known about the universe. This book investigates what cannot be known. Rather than exploring the amazing facts that science, mathematics, and reason have revealed to us, this work studies what science, mathematics, and reason tell us cannot be revealed. In The Outer Limits of Reason, Noson Yanofsky considers what cannot be predicted, described, or known, and what will never be understood. He discusses the limitations of computers, physics, logic, and our own thought processes.Yanofsky describes simple tasks that would take computers trillions of centuries to complete and other problems that computers can never solve; perfectly formed English sentences that make no sense; different levels of infinity; the bizarre world of the quantum; the relevance of relativity theory; the causes of chaos theory; math problems that cannot be solved by normal means; and statements that are true but cannot be proven. He explains the limitations of our intuitions about the world—our ideas about space, time, and motion, and the complex relationship between the knower and the known.Moving from the concrete to the abstract, from problems of everyday language to straightforward philosophical questions to the formalities of physics and mathematics, Yanofsky demonstrates a myriad of unsolvable problems and paradoxes. Exploring the various limitations of our knowledge, he shows that many of these limitations have a similar pattern and that by investigating these patterns, we can better understand the structure and limitations of reason itself. Yanofsky even attempts to look beyond the borders of reason to see what, if anything, is out there.Noson S. Yanofsky is Professor in the Department of Computer and Information Science at Brooklyn College and The Graduate Center of the City University of New York. He is a coauthor of Quantum Computing for Computer Scientists. Learn more about your ad choices. Visit megaphone.fm/adchoicesSupport our show by becoming a premium member! https://newbooksnetwork.supportingcast.fm/mathematics

Why are girls discouraged from doing science? Why do so many promising women leave science in early and mid-career? Why do women not prosper in the scientific workforce?Not Just for the Boys: Why We Need More Women in Science (Oxford UP, 2023) looks back at how society has historically excluded women from the scientific sphere and discourse, what progress has been made, and how more is still needed. Athene Donald, herself a distinguished physicist, explores societal expectations during both childhood and working life using evidence of the systemic disadvantages women operate under, from the developing science of how our brains are―and more importantly aren't―gendered, to social science evidence around attitudes towards girls and women doing science.It also discusses how science is done in practice, in order to dispel common myths: for example, the perception that science is not creative, or that it is carried out by a lone genius in an ivory tower, myths that can be very off-putting to many sections of the population. A better appreciation of the collaborative, creative, and multi-disciplinary nature of science is likely to lead to its appeal to a far wider swathe of people, especially women. This book examines the modern way of working in scientific research, and how gender bias operates in various ways within it, drawing on the voices of leading women in science describing their feelings and experiences. It argues the moral and business case for greater diversity in modern research, the better to improve science and tackle the great challenges we face today.Athene Donald is Professor Emerita in Experimental Physics and Master of Churchill College, University of Cambridge. Other than four years postdoctoral research in the USA, she has spent her career in Cambridge, specializing in soft matter physics and physics at the interface with biology. She was the University of Cambridge's first Gender Equality Champion, and has been involved in numerous initiatives concerning women in science. She was elected Fellow of the Royal Society in 1999 and appointed DBE for services to Physics in 2010.Morteza Hajizadeh is a Ph.D. graduate in English from the University of Auckland in New Zealand. His research interests are Cultural Studies; Critical Theory; Environmental History; Medieval (Intellectual) History; Gothic Studies; 18th and 19th Century British Literature. YouTube channel. Learn more about your ad choices. Visit megaphone.fm/adchoicesSupport our show by becoming a premium member! https://newbooksnetwork.supportingcast.fm/mathematics