A brilliant tour of mathematical thought and a guide to becoming a better thinker, How Not to Be Wrong shows that math is not just a long list of rules to be learned and carried out by rote. Math touches everything we do; It's what makes the world make sense. Using the mathematician's methods and hard-won insights-minus the jargon-professor and popular columnist Jordan Ellenberg guides general readers through his ideas with rigor and lively irreverence, infusing everything from election results to baseball to the existence of God and the psychology of slime molds with a heightened sense of clarity and wonder. Armed with the tools of mathematics, we can see the hidden structures beneath the messy and chaotic surface of our daily lives. How Not to Be Wrong shows us how--Publisher's description.
In this ambitious study, David Corfield attacks the widely held view that it is the nature of mathematical knowledge which has shaped the way in which mathematics is treated philosophically and claims that contingent factors have brought us to the present thematically limited discipline. Illustrating his discussion with a wealth of examples, he sets out a variety of approaches to new thinking about the philosophy of mathematics, ranging from an exploration of whether computers producing mathematical proofs or conjectures are doing real mathematics, to the use of analogy, the prospects for a Bayesian confirmation theory, the notion of a mathematical research programme and the ways in which new concepts are justified. His inspiring book challenges both philosophers and mathematicians to develop the broadest and richest philosophical resources for work in their disciplines and points clearly to the ways in which this can be done.
Mathematicians like to point out that mathematics is universal. In spite of this, most people continue to view it as either mundane (balancing a checkbook) or mysterious (cryptography). This fifth volume of the What's Happening series contradicts that view by showing that mathematics is indeed found everywhere-in science, art, history, and our everyday lives. Here is some of what you'll find in this volume: Mathematics and Science Mathematical biology: Mathematics was key tocracking the genetic code. Now, new mathematics is needed to understand the three-dimensional structure of the proteins produced from that code. Celestial mechanics and cosmology: New methods have revealed a multitude of solutions to the three-body problem. And other new work may answer one of cosmology'smost fundamental questions: What is the size and shape of the universe? Mathematics and Everyday Life Traffic jams: New models are helping researchers understand where traffic jams come from-and maybe what to do about them! Small worlds: Researchers have found a short distance from theory to applications in the study of small world networks. Elegance in Mathematics Beyond Fermat's Last Theorem: Number theorists are reaching higher ground after Wiles' astounding 1994 proof: new developments inthe elegant world of elliptic curves and modular functions. The Millennium Prize Problems: The Clay Mathematics Institute has offered a million dollars for solutions to seven important and difficult unsolved problems. These are just some of the topics of current interest that are covered in thislatest volume of What's Happening in the Mathematical Sciences. The book has broad appeal for a wide spectrum of mathematicians and scientists, from high school students through advanced-level graduates and researchers.
Physics and mathematics have always been closely intertwined, with developments in one field frequently inspiring the other. Currently, there are many unsolved problems in physics which will likely require innovations in mathematical physics. Mathematical physics is concerned with problems in statistical mechanics, atomic and molecular physics, quantum field theory, and, in general, with the mathematical foundations of theoretical physics. mechanics (both nonrelativistic and relativistic), atomic and molecular physics, the existence and properties of the phases of model ferromagnets, the stability of matter, the theory of symmetry and symmetry breaking in quantum field theory (both in general and in concrete models), and mathematical developments in functional analysis and algebra to which such subjects lead. This book presents leading-edge research in this fast-moving field. Structure of the Kalb-Ramond Gauge Symmetry and Spinor Representations; Group Theoretical Interpretation of CPT-Theorem; Cross Recurrence Plots and Their Applications; Analytical Solutions of the Radiative Transfer Equation in One-dimensional Spherical Geometry With Central Symmetry; Hyperspherical Functions and Harmonic Analysis on the Lorentz Group; The Next Stage: Quantum Game Theory; Index.
"This book addresses the challenges that face science and mathematics education if it is to be relevant to 21st century citizens, as well as the ways that outstanding specialists from several countries around the world think it should deal with those challenges. Starting with the issue of science and mathematics teacher education in a changing world, it moves on to deal with innovative approaches to teaching science and mathematics. It then discusses contemporary issues related to the role played by technology in science and mathematics education, the challenges of the STEM agenda, and ways of making science and mathematics education more inclusive. Finally, it focuses on assessment issues, as the success of science and mathematics education depends at least in part on the purposes for which, and ways in which, students' learning is assessed. There is a worldwide trend towards providing meaningful science and mathematics education to all children for the sake of literacy and numeracy development and a need to produce enough science and technology specialists. This trend and need, coupled with the concern raised by students' disengagement in these two knowledge areas and the role that technology may play in countering it, put increasingly high demands on teachers. As shown in this book, science and mathematics education may offer a unique contribution in developing responsible citizens by fostering skills required in order to assume wider responsibilities and roles, focusing on personal, social and environmental dimensions. For instance, it offers unique insights into how teachers can build on students' complicated and interconnected real-worlds to help them learn authentic and relevant science and mathematics. Additionally, the book highlights potential positive relationships between science and mathematics, which are often envisaged as having a conflicting relationship in school curricula. By uncovering the similarities between them, and by providing evidence that both areas deal with issues that are relevant for citizens' daily lives, the book explores ways of linking and giving coherence to science and mathematics knowledge as components of everyday life settings. It also provides directions for future research on the educational potential of interconnecting science and mathematics at the different educational levels. Therefore, this is a worthwhile book for researchers, teacher educators and schoolteachers. It covers theoretical perspectives, research-based approaches and practical applications that may make a difference in education that is relevant and inclusive for citizens in the 21st century"--
Max Tegmark leads us on an astonishing journey through past, present and future, and through the physics, astronomy and mathematics that are the foundation of his work, most particularly his hypothesis that our physical reality is a mathematical structure and his theory of the ultimate multiverse. In a dazzling combination of both popular and groundbreaking science, he not only helps us grasp his often mind-boggling theories, but he also shares with us some of the often surprising triumphs and disappointments that have shaped his life as a scientist. Fascinating from first to last—this is a book that has already prompted the attention and admiration of some of the most prominent scientists and mathematicians.
Mathematics for Neuroscientists, Second Edition, presents a comprehensive introduction to mathematical and computational methods used in neuroscience to describe and model neural components of the brain from ion channels to single neurons, neural networks and their relation to behavior. The book contains more than 200 figures generated using Matlab code available to the student and scholar. Mathematical concepts are introduced hand in hand with neuroscience, emphasizing the connection between experimental results and theory. - Fully revised material and corrected text - Additional chapters on extracellular potentials, motion detection and neurovascular coupling - Revised selection of exercises with solutions - More than 200 Matlab scripts reproducing the figures as well as a selection of equivalent Python scripts
While theoretical particle physics is an extraordinarily fascinating field, the incredibly fast pace at which it moves along, combined with the huge amount of background information necessary to perform cutting edge research, poses a formidable challenge for graduate students. This book represents the first in a series designed to assist students in the process of transitioning from coursework to research in particle physics. Rather than reading literally dozens of physics and mathematics texts, trying to assimilate the countless ideas, translate notations and perspectives, and see how it all fits together to get a holistic understanding, this series provides a detailed overview of the major mathematical and physical ideas in theoretical particle physics. Ultimately the ideas will be presented in a unified, consistent, holistic picture, where each topic is built firmly on what has come before, and all topics are related in a clear and intuitive way. This introductory text on quantum field theory and particle physics provides both a self-contained and complete introduction to not only the necessary physical ideas, but also a complete introduction to the necessary mathematical tools. Assuming minimal knowledge of undergraduate physics and mathematics, this book lays both the mathematical and physical groundwork with clear, intuitive explanations and plenty of examples. The book then continues with an exposition of the Standard Model of Particle Physics, the theory that currently seems to explain the universe apart from gravity. Furthermore, this book was written as a primer for the more advanced mathematical and physical ideas to come later in this series.
