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| god made the natural numbers: The Story Of Numbers Asok Kumar Mallik, 2017-07-27 '… this could make an ideal end-of-year prize for a high-school student who is fascinated by all aspects of number. The subsections provide ideas and opportunities for mathematical exploration. This book might also be deemed a suitable resource for first-year undergraduates in that, via independent study, it would allow such students to broaden their knowledge of various number-theoretic ideas. I would recommend it for the purposes given above.'The Mathematical GazetteThis book is more than a mathematics textbook. It discusses various kinds of numbers and curious interconnections between them. Without getting into hardcore and difficult mathematical technicalities, the book lucidly introduces all kinds of numbers that mathematicians have created. Interesting anecdotes involving great mathematicians and their marvelous creations are included. The reader will get a glimpse of the thought process behind the invention of new mathematics. Starting from natural numbers, the book discusses integers, real numbers, imaginary and complex numbers and some special numbers like quaternions, dual numbers and p-adic numbers.Real numbers include rational, irrational and transcendental numbers. Iterations on real numbers are shown to throw up some unexpected behavior, which has given rise to the new science of 'Chaos'. Special numbers like e, pi, golden ratio, Euler's constant, Gauss's constant, amongst others, are discussed in great detail.The origin of imaginary numbers and the use of complex numbers constitute the next topic. It is shown why modern mathematics cannot even be imagined without imaginary numbers. Iterations on complex numbers are shown to generate a new mathematical object called 'Fractal', which is ubiquitous in nature. Finally, some very special numbers, not mentioned in the usual textbooks, and their applications, are introduced at an elementary level.The level of mathematics discussed in this book is easily accessible to young adults interested in mathematics, high school students, and adults having some interest in basic mathematics. The book concentrates more on the story than on rigorous mathematics. |
| god made the natural numbers: Is God a Mathematician? Mario Livio, 2010-01-19 Explores the plausibility of mathematical answers to puzzles in the physical world, in an accessible exploration of the lives and thoughts of such figures as Archimedes, Galileo, and Newton. |
| god made the natural numbers: God Created The Integers Stephen Hawking, 2007-03-29 Bestselling author and physicist Stephen Hawking explores the masterpieces of mathematics, 25 landmarks spanning 2,500 years and representing the work of 15 mathematicians, including Augustin Cauchy, Bernard Riemann, and Alan Turing. This extensive anthology allows readers to peer into the mind of genius by providing them with excerpts from the original mathematical proofs and results. It also helps them understand the progression of mathematical thought, and the very foundations of our present-day technologies. Each chapter begins with a biography of the featured mathematician, clearly explaining the significance of the result, followed by the full proof of the work, reproduced from the original publication. |
| god made the natural numbers: From Kant to Hilbert Volume 2 William Bragg Ewald, William Ewald, 1999 This two-volume work brings together a comprehensive selection of mathematical works from the period 1707-1930. During this time the foundations of modern mathematics were laid, and From Kant to Hilbert provides an overview of the foundational work in each of the main branches of mathmeatics with narratives showing how they were linked. Now available as a separate volume. |
| god made the natural numbers: The Art of Mathematics Jerry P. King, 2013-11-11 The beauty of mathematics eludes all but a small, select handful of people. This monumental classic will illuminate the aesthetic delights of mathematics for all to behold. Why should only a tiny aristocracy hold the key to appreciating the elegance of mathematics? Why should intelligent, cultured people, who can easily articulate the brilliance of Shakespeare's imagery, quake at the prospect of deciphering a simple algebraic formula? Jerry King, a mathematics professor and a poet, razes the barriers between a world of two cultures and hands us the tools for appreciating the art and treasures of this elegant discipline. In his fluid, poetic voice, he initiates us into the splendid wonders of the Mathworld. He provides us with an original framework for contemplating mathematics as art. He deepens our ultimate comprehension of art by comparing the beauty of a Rembrandt as well as a Jackson Pollock with the riches to be mined in an elegant proof. Like the great philosophers of the past, Dr. King searches for pure Truth--a quest possible today only in the realm of mathematics. With his infectious enthusiasm, he explains with utmost clarity the intellectually stimulating underpinnings of both pure and applied mathematics. He goes on to decry how our educational system has failed by perfunctorily teaching us mathematics, depriving us of the pillars of beauty upon which mathematics rests. Never before has a book spoken so eloquently to our soul in instilling an appreciation for the grandeur of mathematics. Through Dr. King, the muses of mathematics will no longer sing for others and not for us. The elegant world of mathematics awaits us all to savor. |
| god made the natural numbers: Basic Mathematics Serge Lang, 1988-01 |
| god made the natural numbers: Numbers Albrecht Beutelspacher, 2016-01-14 Orignally published in German: Zahlen: Geschichte, Gesetze, Geheimnisse (Munich: C.H. Beck, 2013). |
| god made the natural numbers: Euler William Dunham, 2022-01-13 Leonhard Euler was one of the most prolific mathematicians that have ever lived. This book examines the huge scope of mathematical areas explored and developed by Euler, which includes number theory, combinatorics, geometry, complex variables and many more. The information known to Euler over 300 years ago is discussed, and many of his advances are reconstructed. Readers will be left in no doubt about the brilliance and pervasive influence of Euler's work. |
| god made the natural numbers: Uncountable David Nirenberg, Ricardo L. Nirenberg, 2024-05-09 Ranging from math to literature to philosophy, Uncountable explains how numbers triumphed as the basis of knowledge—and compromise our sense of humanity. Our knowledge of mathematics has structured much of what we think we know about ourselves as individuals and communities, shaping our psychologies, sociologies, and economies. In pursuit of a more predictable and more controllable cosmos, we have extended mathematical insights and methods to more and more aspects of the world. Today those powers are greater than ever, as computation is applied to virtually every aspect of human activity. Yet, in the process, are we losing sight of the human? When we apply mathematics so broadly, what do we gain and what do we lose, and at what risk to humanity? These are the questions that David and Ricardo L. Nirenberg ask in Uncountable, a provocative account of how numerical relations became the cornerstone of human claims to knowledge, truth, and certainty. There is a limit to these number-based claims, they argue, which they set out to explore. The Nirenbergs, father and son, bring together their backgrounds in math, history, literature, religion, and philosophy, interweaving scientific experiments with readings of poems, setting crises in mathematics alongside world wars, and putting medieval Muslim and Buddhist philosophers in conversation with Einstein, Schrödinger, and other giants of modern physics. The result is a powerful lesson in what counts as knowledge and its deepest implications for how we live our lives. |
| god made the natural numbers: The Foundations of Mathematics in the Theory of Sets John P. Mayberry, 2000 This book presents a unified approach to the foundations of mathematics in the theory of sets, covering both conventional and finitary (constructive) mathematics. It is based on a philosophical, historical and mathematical analysis of the relation between the concepts of 'natural number' and 'set'. The author investigates the logic of quantification over the universe of sets and discusses its role in second order logic, as well as in the analysis of proof by induction and definition by recursion. Suitable for graduate students and researchers in both philosophy and mathematics. |
| god made the natural numbers: Algebra and Number Theory Martyn R. Dixon, Leonid A. Kurdachenko, Igor Ya Subbotin, 2011-07-15 Explore the main algebraic structures and number systems that play a central role across the field of mathematics Algebra and number theory are two powerful branches of modern mathematics at the forefront of current mathematical research, and each plays an increasingly significant role in different branches of mathematics, from geometry and topology to computing and communications. Based on the authors' extensive experience within the field, Algebra and Number Theory has an innovative approach that integrates three disciplines—linear algebra, abstract algebra, and number theory—into one comprehensive and fluid presentation, facilitating a deeper understanding of the topic and improving readers' retention of the main concepts. The book begins with an introduction to the elements of set theory. Next, the authors discuss matrices, determinants, and elements of field theory, including preliminary information related to integers and complex numbers. Subsequent chapters explore key ideas relating to linear algebra such as vector spaces, linear mapping, and bilinear forms. The book explores the development of the main ideas of algebraic structures and concludes with applications of algebraic ideas to number theory. Interesting applications are provided throughout to demonstrate the relevance of the discussed concepts. In addition, chapter exercises allow readers to test their comprehension of the presented material. Algebra and Number Theory is an excellent book for courses on linear algebra, abstract algebra, and number theory at the upper-undergraduate level. It is also a valuable reference for researchers working in different fields of mathematics, computer science, and engineering as well as for individuals preparing for a career in mathematics education. |
| god made the natural numbers: Mathematical Logic Roman Kossak, 2024-04-18 This textbook is a second edition of the successful, Mathematical Logic: On Numbers, Sets, Structures, and Symmetry. It retains the original two parts found in the first edition, while presenting new material in the form of an added third part to the textbook. The textbook offers a slow introduction to mathematical logic, and several basic concepts of model theory, such as first-order definability, types, symmetries, and elementary extensions. Part I, Logic Sets, and Numbers, shows how mathematical logic is used to develop the number structures of classical mathematics. All necessary concepts are introduced exactly as they would be in a course in mathematical logic; but are accompanied by more extensive introductory remarks and examples to motivate formal developments. The second part, Relations, Structures, Geometry, introduces several basic concepts of model theory, such as first-order definability, types, symmetries, and elementary extensions, and shows how they are usedto study and classify mathematical structures. The added Part III to the book is closer to what one finds in standard introductory mathematical textbooks. Definitions, theorems, and proofs that are introduced are still preceded by remarks that motivate the material, but the exposition is more formal, and includes more advanced topics. The focus is on the notion of countable categoricity, which analyzed in detail using examples from the first two parts of the book. This textbook is suitable for graduate students in mathematical logic and set theory and will also be of interest to mathematicians who know the technical aspects of the subject, but are not familiar with its history and philosophical background. |
| god made the natural numbers: Brouwer's Cambridge Lectures on Intuitionism Luitzen Egbertus Jan Brouwer, 1981 Luitzen Egburtus Jan Brouwer founded a school of thought whose aim was to include mathematics within the framework of intuitionistic philosophy; mathematics was to be regarded as an essentially free development of the human mind. What emerged diverged considerably at some points from tradition, but intuitionism has survived well the struggle between contending schools in the foundations of mathematics and exact philosophy. Originally published in 1981, this monograph contains a series of lectures dealing with most of the fundamental topics such as choice sequences, the continuum, the fan theorem, order and well-order. Brouwer's own powerful style is evident throughout the work. |
| god made the natural numbers: A Classical Introduction to Modern Number Theory Kenneth Ireland, Michael Rosen, 2013-04-17 This well-developed, accessible text details the historical development of the subject throughout. It also provides wide-ranging coverage of significant results with comparatively elementary proofs, some of them new. This second edition contains two new chapters that provide a complete proof of the Mordel-Weil theorem for elliptic curves over the rational numbers and an overview of recent progress on the arithmetic of elliptic curves. |
| god made the natural numbers: Making up Numbers: A History of Invention in Mathematics Ekkehard Kopp, 2020-10-23 Making up Numbers: A History of Invention in Mathematics offers a detailed but accessible account of a wide range of mathematical ideas. Starting with elementary concepts, it leads the reader towards aspects of current mathematical research. The book explains how conceptual hurdles in the development of numbers and number systems were overcome in the course of history, from Babylon to Classical Greece, from the Middle Ages to the Renaissance, and so to the nineteenth and twentieth centuries. The narrative moves from the Pythagorean insistence on positive multiples to the gradual acceptance of negative numbers, irrationals and complex numbers as essential tools in quantitative analysis. Within this chronological framework, chapters are organised thematically, covering a variety of topics and contexts: writing and solving equations, geometric construction, coordinates and complex numbers, perceptions of ‘infinity’ and its permissible uses in mathematics, number systems, and evolving views of the role of axioms. Through this approach, the author demonstrates that changes in our understanding of numbers have often relied on the breaking of long-held conventions to make way for new inventions at once providing greater clarity and widening mathematical horizons. Viewed from this historical perspective, mathematical abstraction emerges as neither mysterious nor immutable, but as a contingent, developing human activity. Making up Numbers will be of great interest to undergraduate and A-level students of mathematics, as well as secondary school teachers of the subject. In virtue of its detailed treatment of mathematical ideas, it will be of value to anyone seeking to learn more about the development of the subject. |
| god made the natural numbers: A Journey Through The Realm of Numbers Menny Aka, Manfred Einsiedler, Thomas Ward, 2020-10-03 This book takes the reader on a journey from familiar high school mathematics to undergraduate algebra and number theory. The journey starts with the basic idea that new number systems arise from solving different equations, leading to (abstract) algebra. Along this journey, the reader will be exposed to important ideas of mathematics, and will learn a little about how mathematics is really done. Starting at an elementary level, the book gradually eases the reader into the complexities of higher mathematics; in particular, the formal structure of mathematical writing (definitions, theorems and proofs) is introduced in simple terms. The book covers a range of topics, from the very foundations (numbers, set theory) to basic abstract algebra (groups, rings, fields), driven throughout by the need to understand concrete equations and problems, such as determining which numbers are sums of squares. Some topics usually reserved for a more advanced audience, such as Eisenstein integers or quadratic reciprocity, are lucidly presented in an accessible way. The book also introduces the reader to open source software for computations, to enhance understanding of the material and nurture basic programming skills. For the more adventurous, a number of Outlooks included in the text offer a glimpse of possible mathematical excursions. This book supports readers in transition from high school to university mathematics, and will also benefit university students keen to explore the beginnings of algebraic number theory. It can be read either on its own or as a supporting text for first courses in algebra or number theory, and can also be used for a topics course on Diophantine equations. |
| god made the natural numbers: Two Dozen (or so) Arguments for God Jerry Walls, Trent Dougherty, 2018-08-07 Thirty years ago, Alvin Plantinga gave a lecture called Two Dozen (or so) Theistic Arguments, which served as an underground inspiration for two generations of scholars and students. In it, he proposed a number of novel and creative arguments for the existence of God which have yet to receive the attention they deserve. In Two Dozen (or so) Arguments for God, each of Plantinga's original suggestions, many of which he only briefly sketched, is developed in detail by a wide variety of accomplished scholars. The authors look to metaphysics, epistemology, semantics, ethics, aesthetics, and beyond, finding evidence for God in almost every dimension of reality. Those arguments new to natural theology are more fully developed, and well-known arguments are given new life. Not only does this collection present ground-breaking research, but it lays the foundations for research projects for years to come. |
| god made the natural numbers: A Classical Introduction to Modern Number Theory K. Ireland, M. Rosen, 2013-03-09 This book is a revised and greatly expanded version of our book Elements of Number Theory published in 1972. As with the first book the primary audience we envisage consists of upper level undergraduate mathematics majors and graduate students. We have assumed some familiarity with the material in a standard undergraduate course in abstract algebra. A large portion of Chapters 1-11 can be read even without such background with the aid of a small amount of supplementary reading. The later chapters assume some knowledge of Galois theory, and in Chapters 16 and 18 an acquaintance with the theory of complex variables is necessary. Number theory is an ancient subject and its content is vast. Any intro ductory book must, of necessity, make a very limited selection from the fascinat ing array of possible topics. Our focus is on topics which point in the direction of algebraic number theory and arithmetic algebraic geometry. By a careful selection of subject matter we have found it possible to exposit some rather advanced material without requiring very much in the way oftechnical background. Most of this material is classical in the sense that is was dis covered during the nineteenth century and earlier, but it is also modern because it is intimately related to important research going on at the present time. |
| god made the natural numbers: Discern & Deploy the “Heir” Force Dr. Cecilia Jackson, 2017-10-24 This work focuses on teaching and activating Gods angelic force. It explains the role angels play in the lives of believers and presents understanding of scripture regarding these powerful beings and their assignment to the heirs of salvation. God has given his believers and church prevailing help. This book illuminates the paramount function of angels who are the most powerful force in the supernatural world (other than the Lord Jesus). They work on behalf of believers to help fulfill destiny. Simultaneously, angelic support is available for daily victorious living and for maintaining hope during ongoing challenges. This book expounds on these truths. Discern & Deploy the Heir Force, written for personal study, also includes chapter questions and written assessments for groups. In addition, it is complete with declarations and prayers that activate angelic activity in the lives of believers. |
| god made the natural numbers: Numbers Alfred S. Posamentier, Bernd Thaller, 2015-08-11 Did you grow up thinking math is boring? It’s time to reconsider. This book will teach you everything you ever wondered about numbers—and more. How and why did human beings first start using numbers at the dawn of history? Would numbers exist if we Homo sapiens weren’t around to discover them? What’s so special about weird numbers like pi and the Fibonacci sequence? What about rational, irrational, real, and imaginary numbers? Why do we need them? Two veteran math educators explain it all in ways even the most math phobic will find appealing and understandable. You’ll never look at those squiggles on your calculator the same again. |
| god made the natural numbers: Studying Mathematics Marco Bramanti, Giancarlo Travaglini, 2018-07-23 This book is dedicated to preparing prospective college students for the study of mathematics. It can be used at the end of high school or during the first year of college, for personal study or for introductory courses. It aims to set a meeting between two relatives who rarely speak to each other: the Mathematics of Beauty, which shows up in some popular books and films, and the Mathematics of Toil, which is widely known. Toil can be overcome through an appropriate method of work. Beauty will be found in the achievement of a way of thinking. The first part concerns the mathematical language: the expressions “for all”, “there exists”, “implies”, “is false”, ...; what is a proof by contradiction; how to use indices, sums, induction. The second part tackles specific difficulties: to study a definition, to understand an idea and apply it, to fix a slightly wrong argument, to discuss suggestions, to explain a proof. The third part presents customary techniques and points of view in college mathematics. The reader can choose one of three difficulty levels (A, B, C). |
| god made the natural numbers: Ideas and Their Reception David E. Rowe, John McCleary, 2014-05-10 The History of Modern Mathematics, Volume I: Ideas and their Reception documents the proceedings of the Symposium on the History of Modern Mathematics held at Vassar College in Poughkeepsie, New York on June 20-24, 1989. This book is concerned with the emergence and reception of major ideas in fields that range from foundations and set theory, algebra and invariant theory, and number theory to differential geometry, projective and algebraic geometry, line geometry, and transformation groups. Other topics include the theory of reception for the history of mathematics and British synthetic vs. French analytic styles of algebra in the early American Republic. The early geometrical works of Sophus Lie and Felix Klein, background to Gergonne's treatment of duality, and algebraic geometry in the late 19th century are also elaborated. This volume is intended for students and researchers interested in developments in pure mathematics. |
| god made the natural numbers: Game Theory Aviad Heifetz, 2012-05-31 A guide to the fundamentals of game theory for undergraduates and MBA students. |
| god made the natural numbers: The Logic of Number Neil Tennant, 2022 This book develops Tennant's Natural Logicist account of the foundations of the natural, rational, and real numbers. Tennant uses this framework to distinguish the logical from the intuitive aspects of the basic elements of arithmetic. |
| god made the natural numbers: Language, Brain, and Cognitive Development Jacques Mehler, 2001 The contributions to this collection assess the progress of cognitive science. The questions addressed include: What have we learned or not learned about language, brain, and cognition? Where are we now? Where have we failed? Where have we succeeded? |
| god made the natural numbers: Nature's Numbers Ian Stewart, 2008-08-04 It appears to us that the universe is structured in a deeply mathematical way. Falling bodies fall with predictable accelerations. Eclipses can be accurately forecast centuries in advance. Nuclear power plants generate electricity according to well-known formulas. But those examples are the tip of the iceberg. In Nature's Numbers, Ian Stewart presents many more, each charming in its own way.. Stewart admirably captures compelling and accessible mathematical ideas along with the pleasure of thinking of them. He writes with clarity and precision. Those who enjoy this sort of thing will love this book.—Los Angeles Times |
| god made the natural numbers: Hypernumbers and Extrafunctions Mark Burgin, 2012-05-16 “Hypernumbers and Extrafunctions” presents a rigorous mathematical approach to operate with infinite values. First, concepts of real and complex numbers are expanded to include a new universe of numbers called hypernumbers which includes infinite quantities. This brief extends classical calculus based on real functions by introducing extrafunctions, which generalize not only the concept of a conventional function but also the concept of a distribution. Extrafucntions have been also efficiently used for a rigorous mathematical definition of the Feynman path integral, as well as for solving some problems in probability theory, which is also important for contemporary physics. This book introduces a new theory that includes the theory of distributions as a subtheory, providing more powerful tools for mathematics and its applications. Specifically, it makes it possible to solve PDE for which it is proved that they do not have solutions in distributions. Also illustrated in this text is how this new theory allows the differentiation and integration of any real function. This text can be used for enhancing traditional courses of calculus for undergraduates, as well as for teaching a separate course for graduate students. |
| god made the natural numbers: Excursions in the History of Mathematics Israel Kleiner, 2012-02-02 This book comprises five parts. The first three contain ten historical essays on important topics: number theory, calculus/analysis, and proof, respectively. Part four deals with several historically oriented courses, and Part five provides biographies of five mathematicians who played major roles in the historical events described in the first four parts of the work. Excursions in the History of Mathematics was written with several goals in mind: to arouse mathematics teachers’ interest in the history of their subject; to encourage mathematics teachers with at least some knowledge of the history of mathematics to offer courses with a strong historical component; and to provide an historical perspective on a number of basic topics taught in mathematics courses. |
| god made the natural numbers: Abraham's Dice Karl W. Giberson, 2016-04-18 Most of us believe everything happens for a reason. Whether it is God's will,karma, or fate, we want to believe that nothing in the world, especially disasters and tragedies, is a random, meaningless event. But now, as never before, confident scientific assertions that the world embodies a profound contingency are challenging theological claims that God acts providentially in the world. The random and meandering path of evolution is widely used as an argument that God did not create life. Abraham's Dice explores the interplay between chance and providence in the monotheistic religious traditions, looking at how their interaction has been conceptualized as our understanding of the workings of nature has changed. This lively historical conversation has generated intense ongoing theological debates, and provocative responses from science: what are we to make of the history of our universe, where chance and law have played out in complex ways? Or the evolution of life, where random mutations have challenged attempts to find purpose within evolution and convinced many that human beings are but a glorious accident? The enduring belief that everything happens for a reason is examined through a conversation with major scholars, among them holders of prestigious chairs at Oxford and Cambridge Universities and the University of Basel, as well as several Gifford lecturers, and two Templeton prize winners. Organized historically, Abraham's Dice provides a wide-ranging scientific, theological, and biblical foundation to address the question of providence and divine action in a world shot through with contingency. |
| god made the natural numbers: Elements of Mathematics Gabor Toth, 2021-09-23 This textbook offers a rigorous presentation of mathematics before the advent of calculus. Fundamental concepts in algebra, geometry, and number theory are developed from the foundations of set theory along an elementary, inquiry-driven path. Thought-provoking examples and challenging problems inspired by mathematical contests motivate the theory, while frequent historical asides reveal the story of how the ideas were originally developed. Beginning with a thorough treatment of the natural numbers via Peano’s axioms, the opening chapters focus on establishing the natural, integral, rational, and real number systems. Plane geometry is introduced via Birkhoff’s axioms of metric geometry, and chapters on polynomials traverse arithmetical operations, roots, and factoring multivariate expressions. An elementary classification of conics is given, followed by an in-depth study of rational expressions. Exponential, logarithmic, and trigonometric functions complete the picture, driven by inequalities that compare them with polynomial and rational functions. Axioms and limits underpin the treatment throughout, offering not only powerful tools, but insights into non-trivial connections between topics. Elements of Mathematics is ideal for students seeking a deep and engaging mathematical challenge based on elementary tools. Whether enhancing the early undergraduate curriculum for high achievers, or constructing a reflective senior capstone, instructors will find ample material for enquiring mathematics majors. No formal prerequisites are assumed beyond high school algebra, making the book ideal for mathematics circles and competition preparation. Readers who are more advanced in their mathematical studies will appreciate the interleaving of ideas and illuminating historical details. |
| god made the natural numbers: Cantorian Set Theory and Limitation of Size Michael Hallett, 1986 This volume presents the philosophical and heuristic framework Cantor developed and explores its lasting effect on modern mathematics. Establishes a new plateau for historical comprehension of Cantor's monumental contribution to mathematics. --The American Mathematical Monthly |
| god made the natural numbers: Text Book of Linear Algebra M.R. Adhikari, 2004 |
| god made the natural numbers: "I AM" Mark Glouberman, 2019-01-01 For whom was the Hebrew Bible written? How much truth does it contain? What, according to the Bible, is the place of men and women in the world? What connection is there between the Bible and morality? In I AM Mark Glouberman supplies new answers to these old questions. He does this by establishing that the foundational scripture of the West is, first and foremost, a philosophical document, not a theological tract, nor yet the religious history of a nation. The author identifies the Bible's fundamental principle, the ontological principle of particularity. This principle, he shows, is what makes the Bible the revolutionary text that it is. God's I AM WHO I AM asserts the principle, of which the Bible's deity is a personified form. God's self-identification also points to the real, anthropological, meaning of the ism called monotheism. A portion of Glouberman's book is devoted to illustrating the Bible's live relevance in many of the areas where modern philosophers congregate, including moral philosophy, political philosophy, metaphysics, and epistemology. Isn't it a bit late in the day for the Bible's meaning to be revealed? Glouberman says that it's about time. |
| god made the natural numbers: Operator Theory and Ill-Posed Problems Mikhail M. Lavrent'ev, Lev Ja. Savel'ev, 2011-12-22 This book consists of three major parts. The first two parts deal with general mathematical concepts and certain areas of operator theory. The third part is devoted to ill-posed problems. It can be read independently of the first two parts and presents a good example of applying the methods of calculus and functional analysis. The first part Basic Concepts briefly introduces the language of set theory and concepts of abstract, linear and multilinear algebra. Also introduced are the language of topology and fundamental concepts of calculus: the limit, the differential, and the integral. A special section is devoted to analysis on manifolds. The second part Operators describes the most important function spaces and operator classes for both linear and nonlinear operators. Different kinds of generalized functions and their transformations are considered. Elements of the theory of linear operators are presented. Spectral theory is given a special focus. The third part Ill-Posed Problems is devoted to problems of mathematical physics, integral and operator equations, evolution equations and problems of integral geometry. It also deals with problems of analytic continuation. Detailed coverage of the subjects and numerous examples and exercises make it possible to use the book as a textbook on some areas of calculus and functional analysis. It can also be used as a reference textbook because of the extensive scope and detailed references with comments. |
| god made the natural numbers: Mathematics with Understanding Harold Fletcher, Arnold A. Howell, 2014-05-17 Mathematics with Understanding: Book 2 is an eight-chapter book that begins with an explanation of the number systems. Fractions, rational numbers, systems of integers, algebraic relations, shape, and measurement are then described. The modular arithmetic and groups as well as probability are also explained. This book will be useful to students and teachers for a deeper understanding of the structure behind many mathematical ideas and processes. |
| god made the natural numbers: Algebra & Geometry Mark V. Lawson, 2016-06-21 Algebra & Geometry: An Introduction to University Mathematics provides a bridge between high school and undergraduate mathematics courses on algebra and geometry. The author shows students how mathematics is more than a collection of methods by presenting important ideas and their historical origins throughout the text. He incorporates a hands-on approach to proofs and connects algebra and geometry to various applications. The text focuses on linear equations, polynomial equations, and quadratic forms. The first several chapters cover foundational topics, including the importance of proofs and properties commonly encountered when studying algebra. The remaining chapters form the mathematical core of the book. These chapters explain the solution of different kinds of algebraic equations, the nature of the solutions, and the interplay between geometry and algebra |
| god made the natural numbers: Euclidean Geometry Mark Solomonovich, 2010 This textbook is a self-contained presentation of Euclidean Geometry, a subject that has been a core part of school curriculum for centuries. The discussion is rigorous, axiom-based, written in a traditional manner, true to the Euclidean spirit. Transformations in the Euclidean plane are included as part of the axiomatics and as a tool for solving construction problems. The textbook can be used for teaching a high school or an introductory level college course. It can be especially recommended for schools with enriched mathematical programs and for homeschoolers looking for a rigorous traditional discussion of geometry. The text is supplied with over 1200 questions and problems, ranging from simple to challenging. The solutions sections of the book contain about 200 answers and hints to solutions and over 100 detailed solutions involving proofs and constructions. More solutions and some supplements for teachers are available in the Instructor's Manual, which is issued as a separate book. Book Reviews: 'In terms of presentation, this text is more rigorous than any existing high school textbook that I know of. It is based on a system of axioms that describe incidence, postulate a notion of congruence of line segments, and assume the existence of enough rigid motions (free mobility)... My gut reaction to the book is, wouldn't it be wonderful if American high school students could be exposed to this serious mathematical treatment of elementary geometry, instead of all the junk that is presented to them in existing textbooks. This book makes no concession to the TV-generation of students who want (or is it the publishers who want it for them?) pretty pictures, side bars, puzzles, games, historical references, cartoons, and all those colored images that clutter the pages of a typical modern textbook, while the mathematical content is diluted more and more with each successive edition.' Professor Robin Hartshorne, University of California at Berkeley. 'The textbook Euclidean Geometry by Mark Solomonovich fills a big gap in the plethora of mathematical textbooks - it provides an exposition of classical geometry with emphasis on logic and rigorous proofs... I would be delighted to see this textbook used in Canadian schools in the framework of an improved geometry curriculum. Until this day comes, I highly recommend Euclidean Geometry by Mark Solomonovich to be used in Mathematics Enrichment Programs across Canada and the USA.' Professor Yuly Billig, Carlton University. |
| god made the natural numbers: Mathematics in a Postmodern Age Russell W. Howell, James Bradley, 2001 The discipline of mathematics has not been spared the sweeping critique of postmodernism. Is mathematical theory true for all time, or are mathematical constructs in fact fallible? This fascinating book examines the tensions that have arisen between modern and postmodern views of mathematics, explores alternative theories of mathematical truth, explains why the issues are important, and shows how a Christian perspective makes a difference. Contributors: W. James Bradley William Dembski Russell W. Howell Calvin Jongsma David Klanderman Christopher Menzel Glen VanBrummelen Scott VanderStoep Michael Veatch Paul Zwier |
| god made the natural numbers: Number Systems Sergei Ovchinnikov, 2015-02-26 This book offers a rigorous and coherent introduction to the five basic number systems of mathematics, namely natural numbers, integers, rational numbers, real numbers, and complex numbers. It is a subject that many mathematicians believe should be learned by any student of mathematics including future teachers. The book starts with the development of Peano arithmetic in the first chapter which includes mathematical induction and elements of recursion theory. It proceeds to an examination of integers that also covers rings and ordered integral domains. The presentation of rational numbers includes material on ordered fields and convergence of sequences in these fields. Cauchy and Dedekind completeness properties of the field of real numbers are established, together with some properties of real continuous functions. An elementary proof of the Fundamental Theorem of Algebra is the highest point of the chapter on complex numbers. The great merit of the book lies in its extensive list of exercises following each chapter. These exercises are designed to assist the instructor and to enhance the learning experience of the students. |
| god made the natural numbers: Number Midhat J. Gazalé, 2000 Publisher Description |
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And to the angel of the church in Sardis, write: These things says He Who has the seven spirits of God and the seven stars . . . (Revelation 3:1). And proceeding from the throne were lightnings …
God's Seven Curses - Bible Study
The last of God's seven curses involves Jesus. The Lord, who was God in the flesh, entered Jerusalem with his disciples a few days before his death. Being hungry, and noticing a fig tree …
Meaning of Numbers in the Bible - Bible Study
God is 'The Great Geometrician' and does everything after a plan by number, weight, and measure. "If God is the Author of the Scriptures and the Creator of the Universe (and He is) then the …
Amazing Facts about God! - Bible Study
God has promised not only to forgive our sins but also to exercise his unlimited power and completely erase from his memory all traces of our disobedience (Isaiah 43:25, Hebrews 8:12, …
Who Has God Personally Killed? - Bible Study
God declared, through an unnamed prophet, that he would have the two men (Eli's sons) executed on the same day because of their many sins (1Samuel 2:25, 34). This prophecy was fulfilled when …
Why Did God Want to Kill Moses? - Bible Study
While the Bible does not state all of reasons God used to justify wanting to kill Moses, we can take what information is available and derive a fairly good explanation. Although it may seem …
Who Is God? - Bible Study
God is a personal, all-powerful, all-knowing, eternal, loving, spirit-composed family currently composed of the Father and Jesus Christ (see John 10:30 - 31, 17:20 - 23, 1John 3:1 - 2). The …
What Does God Look Like? - Bible Study
There are plenty of other places in the Bible that reveal the various parts of what God (the Father and Jesus Christ) looks like as a spirit being. God is recorded as possessing a head …
Meaning of the Number 7 in the Bible - Bible Study
Numbers 7, with its 89 verses, is the second largest single chapter in God's word! The biggest is Psalm 119 with a whopping 176 verses. The book of the minor prophet Micah contains seven …
Where Did God Come From? - Bible Study
In the beginning was the Word, and the Word was with God, and the Word was God . . . All things came into being through Him, and not even one thing that was created came into being …
What Are the Seven Spirits of God? - Bible Study
And to the angel of the church in Sardis, write: These things says He Who has the seven spirits of God and the seven stars . . . (Revelation 3:1). And proceeding from the throne were lightnings …
God's Seven Curses - Bible Study
The last of God's seven curses involves Jesus. The Lord, who was God in the flesh, entered Jerusalem with his disciples a few days before his death. Being hungry, and noticing a fig tree …
Meaning of Numbers in the Bible - Bible Study
God is 'The Great Geometrician' and does everything after a plan by number, weight, and measure. "If God is the Author of the Scriptures and the Creator of the Universe (and He is) …
Amazing Facts about God! - Bible Study
God has promised not only to forgive our sins but also to exercise his unlimited power and completely erase from his memory all traces of our disobedience (Isaiah 43:25, Hebrews 8:12, …
Who Has God Personally Killed? - Bible Study
God declared, through an unnamed prophet, that he would have the two men (Eli's sons) executed on the same day because of their many sins (1Samuel 2:25, 34). This prophecy was …
Why Did God Want to Kill Moses? - Bible Study
While the Bible does not state all of reasons God used to justify wanting to kill Moses, we can take what information is available and derive a fairly good explanation. Although it may seem …
