The Big Bang of Numbers : How to Build the Universe Using only Math by 
Kant's Mathematical World : Mathematics, Cognition, and Experience by
Kant's Mathematical World aims to transform our understanding of Kant's philosophy of mathematics and his account of the mathematical character of the world. Daniel Sutherland reconstructs Kant's project of explaining both mathematical cognition and our cognition of the world in terms of our most basic cognitive capacities. He situates Kant in a long mathematical tradition with roots in Euclid's Elements, and thereby recovers the very different way of thinking about mathematics which existed prior to its "arithmetization" in the nineteenth century. He shows that Kant thought of mathematics as a science of magnitudes and their measurement, and all objects of experience as extensive magnitudes whose real properties have intensive magnitudes, thus tying mathematics directly to the world. His book will appeal to anyone interested in Kant's critical philosophy -- either his account of the world of experience, or his philosophy of mathematics, or how the two inform each other.
Computable Structure Theory : Within the Arithmetic by
In mathematics, we know there are some concepts - objects, constructions, structures, proofs - that are more complex and difficult to describe than others. Computable structure theory quantifies and studies the complexity of mathematical structures, structures such as graphs, groups, and orderings. Written by a contemporary expert in the subject, this is the first full monograph on computable structure theory in 20 years. Aimed at graduate students and researchers in mathematical logic, it brings new results of the author together with many older results that were previously scattered across the literature and presents them all in a coherent framework, making it easier for the reader to learn the main results and techniques in the area for application in their own research. This volume focuses on countable structures whose complexity can be measured within arithmetic; a forthcoming second volume will study structures beyond arithmetic.
Fact or Fluke? : A Critical Look at Statistical Evidence by
Statistics is more topical than ever. Numerous decisions depend on statistical considerations: just think of the Corona crisis or decisions about approving new drugs or other products. If researchers announce they have proved some fact using statistical tests, can we then always be sure that their claim is correct? How, and more importantly why, does statistics work? What can we expect from statistics and what not? Fact or Fluke? is not a textbook that explains statistical tests to the reader; instead, it discusses what comes before those tests: the philosophy behind the statistics. Should one carry out tests, or are there other ways to look at statistics? Ronald Meester and Klaas Slooten use a variety of examples - from court cases to theoretical physics - to present different views on statistics and provide arguments for what they think is the best point of view. This book is meant for anyone who is in some way concerned with, or interested in, statistical evidence: scientific researchers, students, teachers, mathematicians, philosophers, lawyers, managers, and no doubt many others.
The Story of Proof : Logic and the History of Mathematics by
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.
Theory of Clean Rings and Matrices by
This is the first monograph devoted to clean ring and matrix theory. It aims to study a theory of expressing an element in a ring as the sum of some special ones, such as idempotents, units, nilpotents, tripotents, involutions, etc. A matrix over such rings is thereby expressed as the sum of some special matrices. Also another topics on the behaviors of topological properties and -properties of such rings are investigated. The book is based on the results of various published papers, particularly, by the authors'. It is accessible for students familiar with general abstract algebra, while the topics are interesting for researchers in the field of ring, matrix and operator theory. Book jacket.
The Man from the Future by
Born in Budapest at the turn of the century, John von Neumann is one of the most influential scientists to have ever lived. A child prodigy, he mastered calculus by the age of eight, and in high school made lasting contributions to mathematics. In Germany, where he helped lay the foundations of quantum mechanics, and later at Princeton, von Neumann's colleagues believed he had the fastest brain on the planet--bar none. He was instrumental in the Manhattan Project and the design of the atom bomb; he helped formulate the bedrock of Cold War geopolitics and modern economic theory; he created the first ever programmable digital computer; he prophesized the potential of nanotechnology; and, from his deathbed he expounded on the limits of brains and computers--and how they might be overcome. Taking us on an astonishing journey, Ananyo Bhattacharya explores how a combination of genius and unique historical circumstance allowed a single man to sweep through a stunningly diverse array of fields, sparking revolutions wherever he went. The Man from the Future is an insightful and thrilling intellectual biography of the visionary thinker who shaped our century.
Classical and Modern Optimization by
The quest for the optimal is ubiquitous in nature and human behavior. The field of mathematical optimization has a long history and remains active today, particularly in the development of machine learning. Classical and Modern Optimization presents a self-contained overview of classical and modern ideas and methods in approaching optimization problems. The approach is rich and flexible enough to address smooth and non-smooth, convex and non-convex, finite or infinite-dimensional, static or dynamic situations. The first chapters of the book are devoted to the classical toolbox: topology and functional analysis, differential calculus, convex analysis and necessary conditions for differentiable constrained optimization. The remaining chapters are dedicated to more specialized topics and applications. Valuable to a wide audience, including students in mathematics, engineers, data scientists or economists, Classical and Modern Optimization contains more than 200 exercises to assist with self-study or for anyone teaching a third- or fourth-year optimization class.
Groups and Topological Dynamics by
This book is devoted to group-theoretic aspects of topological dynamics such as studying groups using their actions on topological spaces, using group theory to study symbolic dynamics, and other connections between group theory and dynamical systems. One of the main applications of this approach to group theory is the study of asymptotic properties of groups such as growth and amenability. The book presents recently developed techniques of studying groups of dynamical origin using the structure of their orbits and associated groupoids of germs, applications of the iterated monodromy groups to hyperbolic dynamical systems, topological full groups and their properties, amenable groups, groups of intermediate growth, and other topics. The book is suitable for graduate students and researchers interested in group theory, transformations defined by automata, topological and holomorphic dynamics, and theory of topological groupoids. Each chapter is supplemented by exercises of various levels of complexity.
The Evolution of Mathematics : A Rhetorical Approach by
There is a growing awareness among researchers in the humanities and social sciences of the rhetorical force of mathematical discourse--whether in regard to gerrymandering, facial recognition technologies, or racial biases in algorithmic automation. This book proposes a novel way to engage with and understand mathematics via a theoretical framework that highlights how math transforms the social-material world. In this study, G. Mitchell Reyes applies contemporary rhetorical analysis to mathematical discourse, calling into question the commonly held view that math equals truth. Examining mathematics in historical context, Reyes traces its development from Plato's teaching about abstract numbers to Euclidian geometry and the emergence of calculus and infinitesimals, imaginary numbers, and algorithms. This history reveals that mathematical innovation has always relied on rhetorical practices of making meaning, such as analogy, metaphor, and invention. Far from expressing truth hidden deep in reality, mathematics is dynamic and evolving, shaping reality and our experience of it. By bringing mathematics back down to the material-social world, Reyes makes it possible for scholars of the rhetoric and sociology of science, technology, and math to collaborate with mathematicians themselves in order to better understand our material world and public culture.
Categorical Data Analysis and Multilevel Modeling Using R by
Categorical Data Analysis and Multilevel Modeling Using R provides a practical guide to regression techniques for analyzing binary, ordinal, nominal, and count response variables using the R software. Author Xing Liu offers a unified framework for both single-level and multilevel modeling of categorical and count response variables with both frequentist and Bayesian approaches. Each chapter demonstrates how to conduct the analysis using R, how to interpret the models, and how to present the results for publication. A companion website for this book contains datasets and R commands used in the book for students, and solutions for the end-of-chapter exercises on the instructor site.
Algebraic Geometry by
This book is an introduction to the geometry of complex algebraic varieties. It is intended for students who have learned algebra, analysis, and topology, as taught in standard undergraduate courses. So it is a suitable text for a beginning graduate course or an advanced undergraduate course. The book begins with a study of plane algebraic curves, then introduces affine and projective varieties, going on to dimension and construcibility. O-modules (quasicoherent sheaves) are defined without reference to sheaf theory, and their cohomology is defined axiomatically. The Riemann-Roch Theorem for curves is proved using projection to the projective line. Some of the points that aren't always treated in beginning courses are Hensel's Lemma, Chevalley's Finiteness Theorem, and the Birkhoff-Grothendieck Theorem. The book contains extensive discussions of finite group actions, lines in P3, and double planes, and it ends with applications of the Riemann-Roch Theorem.
Gibbs Measures in Biology and Physics : The Potts Model by
This book presents recently obtained mathematical results on Gibbs measures of the q-state Potts model on the integer lattice and on Cayley trees. It also illustrates many applications of the Potts model to real-world situations in biology, physics, financial engineering, medicine, and sociology, as well as in some examples of alloy behavior, cell sorting, flocking birds, flowing foams, and image segmentation.Gibbs measure is one of the important measures in various problems of probability theory and statistical mechanics. It is a measure associated with the Hamiltonian of a biological or physical system. Each Gibbs measure gives a state of the system.The main problem for a given Hamiltonian on a countable lattice is to describe all of its possible Gibbs measures. The existence of some values of parameters at which the uniqueness of Gibbs measure switches to non-uniqueness is interpreted as a phase transition.This book informs the reader about what has been (mathematically) done in the theory of Gibbs measures of the Potts model and the numerous applications of the Potts model. The main aim is to facilitate the readers (in mathematical biology, statistical physics, applied mathematics, probability and measure theory) to progress into an in-depth understanding by giving a systematic review of the theory of Gibbs measures of the Potts model and its applications.
