"Inspiring and informative...deserves to be widely read." -Wall Street Journal "This fun book offers a philosophical take on number systems and revels in the beauty of math." -Science News Because we have ten fingers, grouping by ten seems natural, but twelve would be better for divisibility, and eight is well suited to repeated halving. Grouping by two, as in binary code, has turned out to have its own remarkable advantages. Paul Lockhart presents arithmetic not as rote manipulation of numbers-a practical if mundane branch of knowledge best suited for filling out tax forms-but as a fascinating, sometimes surprising intellectual craft that arises from our desire to add, divide, and multiply important things. Passionate and entertaining, Arithmetic invites us to experience the beauty of mathematics through the eyes of a beguiling teacher. "A nuanced understanding of working with numbers, gently connecting procedures that we once learned by rote with intuitions long since muddled by education... Lockhart presents arithmetic as a pleasurable pastime, and describes it as a craft like knitting." -Jonathon Keats, New Scientist "What are numbers, how did they arise, why did our ancestors invent them, and how did they represent them? They are, after all, one of humankind's most brilliant inventions, arguably having greater impact on our lives than the wheel. Lockhart recounts their fascinating story... A wonderful book." -Keith Devlin, author of Finding Fibonacci

The overall topic of the volume, Mathematics for Computation (M4C), is mathematics taking crucially into account the aspect of computation, investigating the interaction of mathematics with computation, bridging the gap between mathematics and computation wherever desirable and possible, and otherwise explaining why not.Recently, abstract mathematics has proved to have more computational content than ever expected. Indeed, the axiomatic method, originally intended to do away with concrete computations, seems to suit surprisingly well the programs-from-proofs paradigm, with abstraction helping not only clarity but also efficiency.Unlike computational mathematics, which rather focusses on objects of computational nature such as algorithms, the scope of M4C generally encompasses all the mathematics, including abstract concepts such as functions. The purpose of M4C actually is a strongly theory-based and therefore, is a more reliable and sustainable approach to actual computation, up to the systematic development of verified software.While M4C is situated within mathematical logic and the related area of theoretical computer science, in principle it involves all branches of mathematics, especially those which prompt computational considerations. In traditional terms, the topics of M4C include proof theory, constructive mathematics, complexity theory, reverse mathematics, type theory, category theory and domain theory.The aim of this volume is to provide a point of reference by presenting up-to-date contributions by some of the most active scholars in each field. A variety of approaches and techniques are represented to give as wide a view as possible and promote cross-fertilization between different styles and traditions.

Alfred Tarski (1901-1983) was a renowned Polish/American mathematician, a giant of the twentieth century, who helped establish the foundations of geometry, set theory, model theory, algebraic logic and universal algebra. Throughout his career, he taught mathematics and logic at universities and sometimes in secondary schools. Many of his writings before 1939 were in Polish and remained inaccessible to most mathematicians and historians until now. This self-contained book focuses on Tarski's early contributions to geometry and mathematics education, including the famous Banach-Tarski paradoxical decomposition of a sphere as well as high-school mathematical topics and pedagogy. These themes are significant since Tarski's later research on geometry and its foundations stemmed in part from his early employment as a high-school mathematics teacher and teacher-trainer. The book contains careful translations and much newly uncovered social background of these works written during Tarski's years in Poland. Alfred Tarski: Early Work in Poland serves the mathematical, educational, philosophical and historical communities by publishing Tarski's early writings in a broadly accessible form, providing background from archival work in Poland and updating Tarski's bibliography. A list of errata can be found on the author Smith's personal webpage.

Since its inception, fuzzy logic has attracted an incredible amount of interest, and this interest continues to grow at an exponential rate. As such, scientists, researchers, educators and practitioners of fuzzy logic continue to expand on the applicability of what and how fuzzy can be utilised in the real-world. In this book, the authors present key application areas where fuzzy has had significant success. The chapters cover a plethora of application domains, proving credence to the versatility and robustness of a fuzzy approach. A better understanding of fuzzy will ultimately allow for a better appreciation of fuzzy. This book provides the reader with a varied range of examples to illustrate what fuzzy logic can be capable of and how it can be applied. The text will be ideal for individuals new to the notion of fuzzy, as well as for early career academics who wish to further expand on their knowledge of fuzzy applications. The book is also suitable as a supporting text for advanced undergraduate and graduate-level modules on fuzzy logic, soft computing, and applications of AI.

The chapters in this timely volume aim to answer the growing interest in Arthur Schopenhauer's logic, mathematics, and philosophy of language by comprehensively exploring his work on mathematical evidence, logic diagrams, and problems of semantics. Thus, this work addresses the lack of research on these subjects in the context of Schopenhauer's oeuvre by exposing their links to modern research areas, such as the "proof without words" movement, analytic philosophy and diagrammatic reasoning, demonstrating its continued relevance to current discourse on logic. Beginning with Schopenhauer's philosophy of language, the chapters examine the individual aspects of his semantics, semiotics, translation theory, language criticism, and communication theory. Additionally, Schopenhauer's anticipation of modern contextualism is analyzed. The second section then addresses his logic, examining proof theory, metalogic, system of natural deduction, conversion theory, logical geometry, and the history of logic. Special focus is given to the role of the Euler diagrams used frequently in his lectures and their significance to broader context of his logic. In the final section, chapters discuss Schopenhauer's philosophy of mathematics while synthesizing all topics from the previous sections, emphasizing the relationship between intuition and concept. Aimed at a variety of academics, including researchers of Schopenhauer, philosophers, historians, logicians, mathematicians, and linguists, this title serves as a unique and vital resource for those interested in expanding their knowledge of Schopenhauer's work as it relates to modern mathematical and logical study.

Harmonic Analysis is an important tool that plays a vital role in many areas of mathematics as well as applications. It studies functions by decomposing them into components that are special functions. A prime example is decomposing a periodic function into a linear combination of sines and cosines. The subject is vast, and this book covers only the selection of topics that was dealt with in the course given at the Courant Institute in 2000 and 2019. These include standard topics like Fourier series and Fourier transforms of functions, as well as issues of convergence of Abel, Feier, and Poisson sums. At a slightly more advanced level the book studies convolutions with singular integrals, fractional derivatives, Sobolev spaces, embedding theorems, Hardy spaces, and BMO. Applications to elliptic partial differential equations and prediction theory are explored. Some space is devoted to harmonic analysis on compact non-Abelian groups and their representations, including some details about two groups: the permutation group and SO(3). The text contains exercises at the end of most chapters and is suitable for advanced undergraduate students as well as first- or second-year graduate students specializing in the areas of analysis, PDE, probability or applied mathematics.

This monograph is devoted to a new class of non-commutative rings, skew Poincare-Birkhoff-Witt (PBW) extensions. Beginning with the basic definitions and ring-module theoretic/homological properties, it goes on to investigate finitely generated projective modules over skew PBW extensions from a matrix point of view. To make this theory constructive, the theory of Groebner bases of left (right) ideals and modules for bijective skew PBW extensions is developed. For example, syzygies and the Ext and Tor modules over these rings are computed. Finally, applications to some key topics in the noncommutative algebraic geometry of quantum algebras are given, including an investigation of semi-graded Koszul algebras and semi-graded Artin-Schelter regular algebras, and the noncommutative Zariski cancellation problem. The book is addressed to researchers in noncommutative algebra and algebraic geometry as well as to graduate students and advanced undergraduate students.

This book is the first systematic treatment of this area so far scattered in a vast number of articles. As in classical topology, concrete problems require restricting the (generalized point-free) spaces by various conditions playing the roles of classical separation axioms. These are typically formulated in the language of points; but in the point-free context one has either suitable translations, parallels, or satisfactory replacements. The interrelations of separation type conditions, their merits, advantages and disadvantages, and consequences are discussed. Highlights of the book include a treatment of the merits and consequences of subfitness, various approaches to the Hausdorff's axiom, and normality type axioms. Global treatment of the separation conditions put them in a new perspective, and, a.o., gave some of them unexpected importance. The text contains a lot of quite recent results; the reader will see the directions the area is taking, and may find inspiration for her/his further work. The book will be of use for researchers already active in the area, but also for those interested in this growing field (sometimes even penetrating into some parts of theoretical computer science), for graduate and PhD students, and others. For the reader's convenience, the text is supplemented with an Appendix containing necessary background on posets, frames and locales.

This edited book focuses on concepts and their applications using the theory of conceptual spaces, one of today's most central tracks of cognitive science discourse. It features 15 papers based on topics presented at the Conceptual Spaces @ Work 2016 conference. The contributors interweave both theory and applications in their papers. Among the first mentioned are studies on metatheories, logical and systemic implications of the theory, as well as relations between concepts and language. Examples of the latter include explanatory models of paradigm shifts and evolution in science as well as dilemmas and issues of health, ethics, and education. The theory of conceptual spaces overcomes many translational issues between academic theoretization and practical applications. The paradigm is mainly associated with structural explanations, such as categorization and meronomy. However, the community has also been relating it to relations, functions, and systems. The book presents work that provides a geometric model for the representation of human conceptual knowledge that bridges the symbolic and the sub-conceptual levels of representation. The model has already proven to have a broad range of applicability beyond cognitive science and even across a number of disciplines related to concepts and representation.

Ernst Zermelo (1871-1953) is regarded as the founder of axiomatic set theory and best-known for the first formulation of the axiom of choice. However, his papers include also pioneering work in applied mathematics and mathematical physics.

This edition of his collected papers will consist of two volumes. Besides providing a biography, the present Volume I covers set theory, the foundations of mathematics, and pure mathematics and is supplemented by selected items from his Nachlass and part of his translations of Homer's Odyssey. Volume II will contain his work in the calculus of variations, applied mathematics, and physics.

The papers are each presented in their original language together with an English translation, the versions facing each other on opposite pages. Each paper or coherent group of papers is preceded by an introductory note provided by an acknowledged expert in the field which comments on the historical background, motivations, accomplishments, and influence.

Quantum mechanics is arguably one of the most successful scientific theories ever and its applications to chemistry, optics, and information theory are innumerable. This book provides the reader with a rigorous treatment of the main mathematical tools from harmonic analysis which play an essential role in the modern formulation of quantum mechanics. This allows us at the same time to suggest some new ideas and methods, with a special focus on topics such as the Wigner phase space formalism and its applications to the theory of the density operator and its entanglement properties. This book can be used with profit by advanced undergraduate students in mathematics and physics, as well as by confirmed researchers.

Kurt Goedel (1906-1978) shook the mathematical world in 1931 by a result that has become an icon of 20th century science: The search for rigour in proving mathematical theorems had led to the formalization of mathematical proofs, to the extent that such proving could be reduced to the application of a few mechanical rules. Goedel showed that whenever the part of mathematics under formalization contains elementary arithmetic, there will be arithmetical statements that should be formally provable but aren't. The result is known as Goedel's first incompleteness theorem, so called because there is a second incompleteness result, embodied in his answer to the question "Can mathematics be proved consistent?" This book offers the first examination of Goedel's preserved notebooks from 1930, written in a long-forgotten German shorthand, that show his way to the results: his first ideas, how they evolved, and how the jewel-like final presentation in his famous publication On formally undecidable propositions was composed.The book also contains the original version of Goedel's incompleteness article, as handed in for publication with no mentioning of the second incompleteness theorem, as well as six contemporary lectures and seminars Goedel gave between 1931 and 1934 in Austria, Germany, and the United States. The lectures are masterpieces of accessible presentations of deep scientific results, readable even for those without special mathematical training, and published here for the first time.

Fuzzy set and logic theory suggest that all natural language linguistic expressions are imprecise and must be assessed as a matter of degree. But in general membership degree is an imprecise notion which requires that Type 2 membership degrees be considered in most applications related to human decision making schemas. Even if the membership functions are restricted to be Type1, their combinations generate an interval - valued Type 2 membership. This is part of the general result that Classical equivalences breakdown in Fuzzy theory. Thus all classical formulas must be reassessed with an upper and lower expression that are generated by the breakdown of classical formulas.

Key features:

- Ontological grounding
- Epistemological justification
- Measurement of Membership
- Breakdown of equivalences
- FDCF is not equivalent to FCCF
- Fuzzy Beliefs
- Meta-Linguistic axioms
- Ontological grounding
- Epistemological justification
- Measurement of Membership
- Breakdown of equivalences
- FDCF is not equivalent to FCCF
- Fuzzy Beliefs
- Meta-Linguistic axioms

This book is a collection of contributions honouring Arnon Avron's seminal work on the semantics and proof theory of non-classical logics. It includes presentations of advanced work by some of the most esteemed scholars working on semantic and proof-theoretical aspects of computer science logic. Topics in this book include frameworks for paraconsistent reasoning, foundations of relevance logics, analysis and characterizations of modal logics and fuzzy logics, hypersequent calculi and their properties, non-deterministic semantics, algebraic structures for many-valued logics, and representations of the mechanization of mathematics. Avron's foundational and pioneering contributions have been widely acknowledged and adopted by the scientific community. His research interests are very broad, spanning over proof theory, automated reasoning, non-classical logics, foundations of mathematics, and applications of logic in computer science and artificial intelligence. This is clearly reflected by the diversity of topics discussed in the chapters included in this book, all of which directly relate to Avron's past and present works. This book is of interest to computer scientists and scholars of formal logic.

This book collects and coherently presents the research that has been undertaken since the author's previous book Module Theory (1998). In addition to some of the key results since 1995, it also discusses the development of much of the supporting material. In the twenty years following the publication of the Camps-Dicks theorem, the work of Facchini, Herbera, Shamsuddin, Puninski, Prihoda and others has established the study of serial modules and modules with semilocal endomorphism rings as one of the promising directions for module-theoretic research. Providing readers with insights into the directions in which the research in this field is moving, as well as a better understanding of how it interacts with other research areas, the book appeals to undergraduates and graduate students as well as researchers interested in algebra.

This volume provides an account of the current state of the theory of combinatory spaces and discusses various applications. Here the term "combinatory space" can be regarded as a system for functional programming and bears no close connection with combinatory logic. The main chapter is divided into three chapters. Chapter 1 deals with computational structures and computability; Chapter 2 considers combinatory spaces; and Chapter 3 embraces computability in iterative combinatory spaces. A number of appendices treats a survey of examples of combinatory spaces. All sections of the chapters contain exercises together with hints for solution where appropriate. For the reading of some parts of the book a knowledge of mathematical logic and recursive function theory would be desirable. The text is mainly aimed at researchers and specialists of mathematical logic and its applications, as well as theoretical computer scientists.

This proceedings volume documents the contributions presented at the conference held at Fairfield University and at the Graduate Center, CUNY in 2018 celebrating the New York Group Theory Seminar, in memoriam Gilbert Baumslag, and to honor Benjamin Fine and Anthony Gaglione. It includes several expert contributions by leading figures in the group theory community and provides a valuable source of information on recent research developments.

MATRIX is Australia's international, residential mathematical research institute. It facilitates new collaborations and mathematical advances through intensive residential research programs, each lasting 1-4 weeks. This book is a scientific record of the five programs held at MATRIX in its first year, 2016: - Higher Structures in Geometry and Physics - Winter of Disconnectedness - Approximation and Optimisation - Refining C*-Algebraic Invariants for Dynamics using KK-theory - Interactions between Topological Recursion, Modularity, Quantum Invariants and Low- dimensional Topology The MATRIX Scientific Committee selected these programs based on their scientific excellence and the participation rate of high-profile international participants. Each program included ample unstructured time to encourage collaborative research; some of the longer programs also included an embedded conference or lecture series. The articles are grouped into peer-reviewed contributions and other contributions. The peer-reviewed articles present original results or reviews on selected topics related to the MATRIX program; the remaining contributions are predominantly lecture notes based on talks or activities at MATRIX.

Discusses in detail a World Formula, which is the unification of the greatest theories in physics, namely quantum theory and Einstein's general theory Demystifies David Hilbert's World Formula by simplifying the complex math involved in it Explains why nobody had realized Hilbert's immortal stroke of genius As a "Theory of Everything" approach, it automatically provides just the most holistic tools for each and every optimization, decision-making or solution-finding problem there can possibly be-be it in physics, social science, medicine, socioeconomy and politics, real or artificial intelligence or, rather generally, philosophy

The Bittinger Worktext Series changed the face of developmental education with the introduction of objective-based worktexts that presented math one concept at a time. This approach allowed students to understand the rationale behind each concept before practicing the associated skills and then moving on to the next topic. With this revision, Marv Bittinger continues to focus on building success through conceptual understanding, while also supporting students with quality applications, exercises, and new review and study materials to help them apply and retain their knowledge.