While we are commonly told that the distinctive method of mathematics is rigorous proof, and that the special topic of mathematics is abstract structure, there has been no agreement among mathematicians, logicians, or philosophers as to just what either of these assertions means. John P. Burgess clarifies the nature of mathematical rigor and of mathematical structure, and above all of the relation between the two, taking into account some of the latest developments in mathematics, including the rise of experimental mathematics on the one hand and computerized formal proofs on the other hand. The main theses of Rigor and Structure are that the features of mathematical practice that a large group of philosophers of mathematics, the structuralists, have attributed to the peculiar nature of mathematical objects are better explained in a different way, as artefacts of the manner in which the ancient ideal of rigor is realized in modern mathematics. Notably, the mathematician must be very careful in deriving new results from the previous literature, but may remain largely indifferent to just how the results in the previous literature were obtained from first principles. Indeed, the working mathematician may remain largely indifferent to just what the first principles are supposed to be, and whether they are set-theoretic or category-theoretic or something else. Along the way to these conclusions, a great many historical developments in mathematics, philosophy, and logic are surveyed. Yet very little in the way of background knowledge on the part of the reader is presupposed.

Designed for novice as well as more experienced researchers, Reason & Rigor by Sharon M. Ravitch and Matthew Riggan presents conceptual frameworks as a mechanism for aligning literature review, research design, and methodology. The book explores the conceptual framework—defined as both a process and a product—that helps to direct and ground researchers as they work through common research challenges. Focusing on published studies on a range of topics and employing both quantitative and qualitative methods, the updated Second Edition features two new chapters and clearly communicates the processes of developing and defining conceptual frameworks.

This is a textbook for a one-term course whose goal is to ease the transition from lower-division calculus courses to upper-division courses in linear and abstract algebra, real and complex analysis, number theory, topology, combinatorics, and so on. Without such a "bridge" course, most upper division instructors feel the need to start their courses with the rudiments of logic, set theory, equivalence relations, and other basic mathematical raw materials before getting on with the subject at hand. Students who are new to higher mathematics are often startled to discover that mathematics is a subject of ideas, and not just formulaic rituals, and that they are now expected to understand and create mathematical proofs. Mastery of an assortment of technical tricks may have carried the students through calculus, but it is no longer a guarantee of academic success. Students need experience in working with abstract ideas at a nontrivial level if they are to achieve the sophisticated blend of knowledge, disci pline, and creativity that we call "mathematical maturity. " I don't believe that "theorem-proving" can be taught any more than "question-answering" can be taught. Nevertheless, I have found that it is possible to guide stu dents gently into the process of mathematical proof in such a way that they become comfortable with the experience and begin asking them selves questions that will lead them in the right direction.

The new edition of a classic text that concentrates on developing general methods for studying the behavior of classical systems, with extensive use of computation. We now know that there is much more to classical mechanics than previously suspected. Derivations of the equations of motion, the focus of traditional presentations of mechanics, are just the beginning. This innovative textbook, now in its second edition, concentrates on developing general methods for studying the behavior of classical systems, whether or not they have a symbolic solution. It focuses on the phenomenon of motion and makes extensive use of computer simulation in its explorations of the topic. It weaves recent discoveries in nonlinear dynamics throughout the text, rather than presenting them as an afterthought. Explorations of phenomena such as the transition to chaos, nonlinear resonances, and resonance overlap to help the student develop appropriate analytic tools for understanding. The book uses computation to constrain notation, to capture and formalize methods, and for simulation and symbolic analysis. The requirement that the computer be able to interpret any expression provides the student with strict and immediate feedback about whether an expression is correctly formulated. This second edition has been updated throughout, with revisions that reflect insights gained by the authors from using the text every year at MIT. In addition, because of substantial software improvements, this edition provides algebraic proofs of more generality than those in the previous edition; this improvement permeates the new edition.

Transform the Way You Lead! Leadership Rigor offers innovation in leadership through its breakthrough approaches for transforming the way you lead. The simple truth is that "how" you lead is the precursor to "what" you can achieve as a leader, yet it is often underestimated, dismissed, or not given a conscious consideration. In March 2014, Bersin by Deloitte published their latest Global Human Capital Trend Survey stating: Building leadership capability is by far the most urgent need for companies today... and companies see the need for leadership at all levels, in all geographies, and across all functional areas. In addition, this continuous need for new and better leaders has accelerated. Leadership Rigor views your development as a journey with a road map rather than a black-box mystery! It is both a practice and a philosophy designed to accelerate your leadership performance and productivity across the life cycle of your career. Already becoming a movement, Leadership Rigor prepares you to lead yourself, teams, and organizations. The essence of Leadership Rigor is creating "change-ready" leaders who can embrace challenges because they have the tools, models, and language to assess, structure, and facilitate aligned actions. They also have the mindset and emotional skills to lean into the change process despite its uncomfortable nature. By innovating on their preparedness first, these "change-ready" leaders are equipped to realize the growth in themselves and in their teams or organizations. Are you ready to take on your personal journey of Leadership Rigor?

Over the past twenty-five years, mathematical concepts associated with geometric phases have come to occupy a central place in our modern understanding of the physics of electrons in solids. These 'Berry phases' describe the global phase acquired by a quantum state as the Hamiltonian is changed. Beginning at an elementary level, this book provides a pedagogical introduction to the important role of Berry phases and curvatures, and outlines their great influence upon many key properties of electrons in solids, including electric polarization, anomalous Hall conductivity, and the nature of the topological insulating state. It focuses on drawing connections between physical concepts and provides a solid framework for their integration, enabling researchers and students to explore and develop links to related fields. Computational examples and exercises throughout provide an added dimension to the book, giving readers the opportunity to explore the central concepts in a practical and engaging way.

A self-contained, mathematical introduction to the driving ideas in equilibrium statistical mechanics, studying important models in detail.