The Story of Proof: Logic and the History of Mathematics
How the concept of proof has enabled the creation of mathematical knowledge The Story of Proof 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. Shedding light on the workings of mathematics at its most fundamental levels, The Story of Proof offers a compelling new perspective on the field's power and progress.
Illustrating Mathematics
This book is for anyone who wishes to illustrate their mathematical ideas, which in our experience means everyone. It is organized by material, rather than by subject area, and purposefully emphasizes the process of creating things, including discussions of failures that occurred along the way. As a result, the reader can learn from the experiences of those who came before, and will be inspired to create their own illustrations. Topics illustrated within include prime numbers, fractals, the Klein bottle, Borromean rings, tilings, space-filling curves, knot theory, billiards, complex dynamics, algebraic surfaces, groups and prime ideals, the Riemann zeta function, quadratic fields, hyperbolic space, and hyperbolic 3-manifolds. Everyone who opens this book should find a type of mathematics with which they identify. Each contributor explains the mathematics behind their illustration at an accessible level, so that all readers can appreciate the beauty of both the object itself and the mathematics behind it.
Mathematical Intelligence: A Story of Human Superiority over Machines
A fresh explorationinto the 'human nature versus technology'argument,revealing anunexpected advantage that humans have over our future robot masters: we're actually good at mathematics. There's so much discussion about the threat posed by intelligent machines that it sometimes seems as though we should simply surrender to our robot overlords now. But Junaid Mubeen isn't ready to throw in the towel just yet. As far as he is concerned, we have the creative edge over computers, because of a remarkable system of thought that humans have developed over the millennia. It's familiar to us all, but often badly taught in schools and misrepresented in popular discourse--math. Computers are, of course, brilliant at totting up sums, pattern-seeking, and performing mindless tasks of, well, computation. For all things calculation, machines reign supreme. But Junaid identifies seven areas of intelligence where humans can retain a crucial edge. And in exploring these areas, he opens up a fascinating world where we can develop our uniquely human mathematical talents. Just a few of the fascinating subjects covered in MATHEMATICAL INTELLIGENCE include: -Humans are endowed with a natural sense of numbers that is based on approximation rather than precise calculation. Our in-builtestimationskills complement the precision of computers. Interpreting the real world depends on both. -What sets humans apart from other animals is language and abstraction. We have an extraordinary ability to create powerfulrepresentationsof knowledge-- more diverse than the binary language of computers. -Mathematics confers the most robust, logical framework for establishing permanent truths.Reasoningshields us from the dubious claims of pure pattern-recognition systems. -All mathematical truths are derived from a starting set of assumptions, or axioms. Unlike computers, humans have the freedom to break free of convention and examine the logical consequences of our choices. Mathematics rewards ourimaginationwith fascinating and, on occasion, applicable concepts that originate from breaking the rules. -Computers can be tasked to solve a range of problems, but which problems are worth the effort?Questioningis as vital to our repertoire of thinking skills as problem-solving itself.
Language and the Rise of the Algorithm
A wide-ranging history of the algorithm. Bringing together the histories of mathematics, computer science, and linguistic thought,Language and the Rise of the Algorithmreveals how recent developments in artificial intelligence are reopening an issue that troubled mathematicians well before the computer age: How do you draw the line between computational rules and the complexities of making systems comprehensible to people? By attending to this question, we come to see that the modern idea of the algorithm is implicated in a long history of attempts to maintain a disciplinary boundary separating technical knowledge from the languages people speak day to day. Here Jeffrey M. Binder offers a compelling tour of four visions of universal computation that addressed this issue in very different ways: G. W. Leibniz's calculus ratiocinator; a universal algebra scheme Nicolas de Condorcet designed during the French Revolution; George Boole's nineteenth-century logic system; and the early programming language ALGOL, short foralgorithmic language.These episodes show that symbolic computation has repeatedly become entangled in debates about the nature of communication. Machine learning, in its increasing dependence on words, erodes the line between technical and everyday language, revealing the urgent stakes underlying this boundary. The idea of the algorithm is a levee holding back the social complexity of language, and it is about to break. This book is about the flood that inspired its construction.
Probably Overthinking It: How to Use Data to Answer Questions, Avoid Statistical Traps, and Make Better Decisions
An essential guide to the ways data can improve decision making. Statistics are everywhere: in news reports, at the doctor's office, and in every sort of forecast, from the stock market to the weather. Blogger, teacher, and computer scientist Allen B. Downey knows well that people have an innate ability both to understand statistics and to be fooled by them. As he makes clear in this accessible introduction to statistical thinking, the stakes are big. Simple misunderstandings have led to incorrect medical prognoses, underestimated the likelihood of large earthquakes, hindered social justice efforts, and resulted in dubious policy decisions. There are right and wrong ways to look at numbers, and Downey will help you see which are which. Probably Overthinking It uses real data to delve into real examples with real consequences, drawing on cases from health campaigns, political movements, chess rankings, and more. He lays out common pitfalls--like the base rate fallacy, length-biased sampling, and Simpson's paradox--and shines a light on what we learn when we interpret data correctly, and what goes wrong when we don't. Using data visualizations instead of equations, he builds understanding from the basics to help you recognize errors, whether in your own thinking or in media reports. Even if you have never studied statistics--or if you have and forgot everything you learned--this book will offer new insight into the methods and measurements that help us understand the world.
Wu Wenjun: A Biography in Pictures
This book depicts the fascinating life story of Wu Wenjun, a renowned mathematician who made significant contribution in the field of topology, ancient Chinese mathematics, and mathematics mechanization. He was a recipient of the Highest Science and Technology Award, the highest scientific award in China, as well as the Shaw Prize in Mathematics.Through vivid illustrations and eloquent writing, this book recounts rarely known anecdotes and significant events from Wu Wenjun's life through his childhood, education, and scientific career, offering insights into his life values.
The SAGE Encyclopedia of Theory in Science, Technology, Engineering, and Mathematics
Project Description: Theories are part and parcel of every human activity that involves knowing about the world and our place in it. In all areas of inquiry from the most commonplace to the most scholarly and esoteric, theorizing plays a fundamental role. The SAGE Encyclopedia of Theory in Science, Technology, Engineering, and Mathematicsfocuses on the ways that various STEM disciplines theorize about their subject matter. How is thinking about the subject organized? What methods are used in moving a novice in given field into the position of a competent student of that subject? Within the pages of this landmark work, readers will learn about the complex decisions that are made when framing a theory, what goes into constructing a powerful theory, why some theories change or fail, how STEM theories reflect socio-historical moments in time and how - at their best - they form the foundations for exploring and unlocking the mysteries of the world around us. Featuring more than 200 authoritative articles written by experts in their respective fields, the encyclopedia includes a Reader's Guide that organizes entries by broad themes; lists of Further Readings and cross-references that conclude each article;and a Resource Guide listing classic books in the field, leading journals, associations, and key websites.
The Art of More: How Mathematics Created Civilization
An illuminating, millennia-spanning history of the impact mathematics has had on the world, and the fascinating people who have mastered its inherent power Counting is not innate to our nature, and without education humans can rarely count past three -- beyond that, it's just "more." But once harnessed by our ancestors, the power of numbers allowed humanity to flourish in ways that continue to lead to discoveries and enrich our lives today. Ancient tax collectors used basic numeracy to fuel the growth of early civilization, navigators used clever geometrical tricks to engage in trade and connect people across vast distances, astronomers used logarithms to unlock the secrets of the heavens, and their descendants put them to use to land us on the moon. In every case, mathematics has proved to be a greatly underappreciated engine of human progress. In this captivating, sweeping history, Michael Brooks acts as our guide through the ages. He makes the case that mathematics was one of the foundational innovations that catapulted humanity from a nomadic existence to civilization, and that it has since then been instrumental in every great leap of humankind. Here are ancient Egyptian priests, Babylonian bureaucrats, medieval architects, dueling Swiss brothers, renaissance painters, and an eccentric professor who invented the infrastructure of the online world. Their stories clearly demonstrate that the invention of mathematics was every bit as important to the human species as was the discovery of fire. From first page to last, The Art of More brings mathematics back into the heart of what it means to be human.
Self-Similar and Self-Affine Sets and Measures
Although there is no precise definition of a ""fractal"", it is usually understood to be a set whose smaller parts, when magnified, resemble the whole. Self-similar and self-affine sets are those for which this resemblance is precise and given by a contracting similitude or affine transformation. The present book is devoted to this most basic class of fractal objects. The book contains both introductory material for beginners and more advanced topics, which continue to be the focus of active research. Among the latter are self-similar sets and measures with overlaps, including the much-studied infinite Bernoulli convolutions. Self-affine systems pose additional challenges; their study is often based on ergodic theory and dynamical systems methods. In the last twenty years there have been many breakthroughs in these fields, and our aim is to give introduction to some of them, often in the simplest nontrivial cases. The book is intended for a wide audience of mathematicians interested in fractal geometry, including students. Parts of the book can be used for graduate and even advanced undergraduate courses.
A Brief History of Mathematics: A Promenade Through the Civilizations of Our World
This volume, originally published in China and translated into four other languages, presents a fascinating and unique account of the history of mathematics, divided into eight chronologically organized chapters. Tracing the development of mathematics across disparate regions and peoples, with particular emphasis on the relationship between mathematics and civilization, it examines mathematical sources and inspirations leading from Egypt, Babylon and ancient Greece and expanding to include Chinese, Indian and Arabic mathematics, the European Renaissance andthe French revolution up through the Nineteenth and Twentieth Centuries. Each chapter explores connections among mathematics and cultural elements of the time and place treated, accompanying the reader in a varied and exciting journey through human civilizations. The book contemplates the intersections of mathematics with other disciplines, including the relationship between modern mathematics and modern art, and the resulting applications, with the aid of images and photographs, often taken by the author, which further enhance the enjoyment for the reader. Written for a general audience, this book will be of interest to anyone who's studied mathematics in university or even high school, while also benefiting researchers in mathematics and the humanities.
Weapons of Math Destruction: How Big Data Increases Inequality and Threatens Democracy
Longlisted for the National Book Award| New York Times Bestseller A former Wall Street quant sounds an alarm on the mathematical models that pervade modern life and threaten to rip apart our social fabric. We live in the age of the algorithm. Increasingly, the decisions that affect our lives--where we go to school, whether we get a car loan, how much we pay for health insurance--are being made not by humans, but by mathematical models. In theory, this should lead to greater fairness: Everyone is judged according to the same rules, and bias is eliminated. But as Cathy O'Neil reveals in this urgent and necessary book, the opposite is true. The models being used today are opaque, unregulated, and uncontestable, even when they're wrong. Most troubling, they reinforce discrimination: If a poor student can't get a loan because a lending model deems him too risky (by virtue of his zip code), he's then cut off from the kind of education that could pull him out of poverty, and a vicious spiral ensues. Models are propping up the lucky and punishing the downtrodden, creating a "toxic co*cktail for democracy." Welcome to the dark side of Big Data. Tracing the arc of a person's life, O'Neil exposes the black box models that shape our future, both as individuals and as a society. These "weapons of math destruction" score teachers and students, sort résumés, grant (or deny) loans, evaluate workers, target voters, set parole, and monitor our health. O'Neil calls on modelers to take more responsibility for their algorithms and on policy makers to regulate their use. But in the end, it's up to us to become more savvy about the models that govern our lives. This important book empowers us to ask the tough questions, uncover the truth, and demand change.
The Lost Scientists of World War II
This book tells the stories of scientists from Germany and other European countries who vanished during World War II. These erudite scholars contributed to diverse scientific fields and were associated with some of the world's leading universities and research institutions. Despite their proficiency, they all sought help from agencies to relocate to the UK in the 1930s, but were unable to secure the necessary assistance.The Lost Scientists of World War II explores the fascinating narratives of thirty of these scientific refugees, delving into the reasons behind the unavailability of aid and presenting fresh insights into the tragic fates or astounding survival experiences of these individuals.
The Secret Lives of Numbers: A Hidden History of Math's Unsung Trailblazers
A new history of mathematics focusing on the marginalized voices who propelled the discipline, spanning six continents and thousands of years of untold stories. "A book to make you love math." --Financial Times Mathematics shapes almost everything we do. But despite its reputation as the study of fundamental truths, the stories we have been told about it are wrong--warped like the sixteenth-century map that enlarged Europe at the expense of Africa, Asia and the Americas.In The Secret Lives of Numbers, renowned math historian Kate Kitagawa and journalist Timothy Revell make the case that the history of math is infinitely deeper, broader, and richer than the narrative we think we know. Our story takes us from Hypatia, the first great female mathematician, whose ideas revolutionized geometry and who was killed for them--to Karen Uhlenbeck, the first woman to win the Abel Prize, "math's Nobel." Along the way we travel the globe to meet the brilliant Arabic scholars of the "House of Wisdom," a math temple whose destruction in the Siege of Baghdad in the thirteenthcentury was a loss arguably on par with that of the Library of Alexandria; Madhava of Sangamagrama, the fourteenth-century Indian genius who uncovered the central tenets of calculus 300 years before Isaac Newton was born; and the Black mathematicians of the Civil Rights era, who played a significant role in dismantling early data-based methods of racial discrimination. Covering thousands of years, six continents, and just about every mathematical discipline,The Secret Lives of Numbersis an immensely compelling narrative history.
Visual Complex Analysis: 25th Anniversary Edition
Complex Analysis is the powerful fusion of the complex numbers (involving the 'imaginary' square root of -1) with ordinary calculus, resulting in a tool that has been of central importance to science for more than 200 years. This book brings this majestic and powerful subject to life by consistently using geometry (not calculation) as the means of explanation. The 501 diagrams of the original edition embodied geometrical arguments that (for the first time) replaced the long and often opaque computations of the standard approach, in force for the previous 200 years, providing direct, intuitive, visual access to the underlying mathematical reality. This new 25th Anniversary Edition introduces brand-new captions that fully explain the geometrical reasoning, making it possible to read the work in an entirely new way--as a highbrow comic book!
Sharing Our Science: How to Write and Speak STEM
A personal, practical, and inspirational guide to written and oral STEM communications for scientists and technical professionals. In Sharing Our Science, scientist-turned-writing teacher Brandon Brown offers an eminently useful guidebook for STEM practitioners looking to communicate their technical work to either a technical or a broader audience. Professionals are increasingly required to communicate their work through blogs, podcasts, and newsletters and to submit to traditional media. After seeing his colleagues struggle to find a writing guide that tackled the unique challenges of writing and speaking about scientific topics, Brown set out to write the definitive handbook to assist STEM students, scientists, engineers, and tech workers alike. In this practical and relevant book, Brown uses his experience as a proven science communicator to cover three levels of writing: fundamental craft considerations, such as narrative tension, structure, sentences, and audience; unique scientific considerations, such as conveying numbers and utilizing metaphors; and finally, social considerations, such as public speaking and writing inside and outside of silos. In place of a reference manual, Brown's engaging narrative guide clarifies the fundamental principles that impact all scientific communication tasks, from white papers and slide decks to Zoom meetings and emails. Sharing Our Science represents the culmination of a lifetime of writing, research, and teaching that will enrich scientists' careers and illuminate the ways in which science is done and conveyed to the world.
Being Human in STEM: Partnering with Students to Shape Inclusive Practices and Communities
For all STEM faculty, chairs, administrators, and faculty developers who work to support students' learning and thriving in STEM - especially those students who have felt unwelcome and unsupported in their past STEM experiences - this book offers sustainable strategies that are now being widely adopted to create inclusive environments in undergraduate STEM classes and programs. Further, this book presents a framework for partnering with students to collaboratively envision how STEM can be a space that fosters a sense of belonging for, and promotes the success of, all individuals in STEM. This book presents the Being Human in STEM Initiative, or HSTEM, as a model for challenging the assumptions we make, and how we communicate to students, about who belongs and who can thrive in STEM. This work arose out of a time of conflict at Amherst College: A four-day sit-in, protesting in support of the Black Lives Matter movement and bringing attention to related experiences of exclusion and marginalization that minoritized students experienced on campus. What emerged from that conflict has been transformative for the college, its students, and for its faculty and staff. In this book, the authors share how the HSTEM course came into being, offer a course overview, readings, and resources for developing an HSTEM course at your own institution, provide recommendations for evaluating the multi-level impact of inclusive change initiatives, and profile models of how the HSTEM course has been adapted at colleges and universities across the country. In addition to providing a road map for developing your own HSTEM course, the authors articulate ways that you can make any course or institutional structure more inclusive through active listening and validation, and through reflective practice and partnership, to progressively make incremental and sustainable changes in STEM education. Through listening and reflecting, the model facilitates uncovering the disconnects that can impede inclusivity in our classrooms and laboratories. While the authors offer a proven process and model for change, originally motivated by the urgent need to respond to students' demands, they recognize that larger institutional culture shifts require the identification and commitment to common values, a shared sense of purpose in the work of change, and the provision of agency and resources to individuals tasked with making change happen. How might we shift institutional STEM culture? The HSTEM model provides one solution: By reflecting on our own lived experiences and identities, engaging with the literature on the factors that enhance and limit full inclusion in STEM, and partnering with students to identify actionable ways to bring about sustainable change in our scientific communities, we can all work towards creating a more inclusive, and human, STEM ecosystem.Each chapter opens with a set of guiding reflective questions to help you connect these ideas, frameworks, and strategies to your own teaching and institutional context. While each chapter builds on the previous ideas and frameworks, the book can also be used as a resource to identify a just-in-time strategy to address particular questions you may have about making your teaching more inclusive. The appendices offer an array of Facilitator Guides, each of which outlines a student-endorsed exercise, based on the pedagogical literature, that can foster a sense of belonging and inclusion in your classrooms and laboratory spaces.
How to Derive a Formula
Will artificial intelligence solve all problems, making scientific formulae redundant? The authors of this book would argue that there is still a vital role in formulating them to make sense of the laws of nature. To derive a formula one needs to follow a series of steps; last of all, check that the result is correct, primarily through the analysis of limiting cases. The book is about unravelling this machinery.Mathematics is the 'queen of all sciences', but students encounter many obstacles in learning the subject -- familiarization with the proofs of hundreds of theorems, mysterious symbols, and technical routines for which the usefulness is not obvious upfront. Those interested in the physical sciences could lose motivation, not seeing the wood for the trees.How to Derive a Formula is an attempt to engage these learners, presenting mathematical methods in simple terms, with more of an emphasis on skills as opposed to technical knowledge. Based on intuition and common sense rather than mathematical rigor, it teaches students from scratch using pertinent examples, many taken across the physical sciences. This book provides an interesting new perspective of what a mathematics textbook could be, including historical facts and humour to complement the material.
Modern Mathematical Logic
This textbook gives a complete and modern introduction to mathematical logic. The author uses contemporary notation, conventions, and perspectives throughout, and emphasizes interactions with the rest of mathematics. In addition to covering the basic concepts of mathematical logic and the fundamental material on completeness, compactness, and incompleteness, it devotes significant space to thorough introductions to the pillars of the modern subject: model theory, set theory, and computability. Requiring only a modest background of undergraduate mathematics, the text can be readily adapted for a variety of one- or two-semester courses at the upper-undergraduate or beginning-graduate level. Numerous examples reinforce the key ideas and illustrate their applications, and a wealth of classroom-tested exercises serve to consolidate readers' understanding. Comprehensive and engaging, this book offers a fresh approach to this enduringly fascinating and important subject.
