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Foundations of Higher Mathematics: A Journey into Abstract Thought
Embarking on the study of higher mathematics can feel like stepping onto a different planet. Gone are the familiar landscapes of arithmetic and algebra, replaced by abstract concepts and rigorous proofs. This journey, however, is profoundly rewarding, unlocking a deeper understanding of the universe and sharpening analytical skills applicable far beyond the academic realm. This comprehensive guide explores the crucial foundations of higher mathematics, providing a roadmap for navigating this fascinating and challenging field. We'll delve into the essential building blocks, illuminating the path towards mastering advanced mathematical concepts.
Understanding the Building Blocks: Logic and Set Theory
Before tackling advanced topics, a firm grasp of fundamental principles is paramount. The very foundation of higher mathematics rests upon two pillars: logic and set theory.
#### Logic: The Language of Mathematics
Logic provides the rigorous framework for mathematical reasoning. It's the language in which mathematical statements are formulated and proofs are constructed. Understanding propositional logic (statements and their truth values), predicate logic (quantifiers like "for all" and "there exists"), and the rules of inference (deductive reasoning) is crucial for building sound mathematical arguments. Without a solid foundation in logic, even the simplest mathematical proofs can become elusive.
#### Set Theory: The Foundation of Mathematical Objects
Set theory, developed by Georg Cantor, provides the language for describing mathematical objects. A set is simply a collection of well-defined objects, and understanding concepts like subsets, unions, intersections, and power sets is essential for working with virtually any mathematical structure. Set theory underpins much of modern mathematics, providing a unifying framework for diverse areas like algebra, topology, and analysis. Understanding different types of sets (finite, infinite, countable, uncountable) is particularly vital as we progress to more advanced topics.
Essential Branches of Higher Mathematics: A Glimpse into the Landscape
Building upon logic and set theory, several crucial branches of higher mathematics form the bedrock for further study.
#### Real Analysis: The Foundation of Calculus
Real analysis rigorously formalizes the concepts of calculus, moving beyond intuitive notions to a precise and axiomatic framework. It delves into the properties of real numbers, limits, continuity, differentiability, and integration, laying the groundwork for understanding more advanced areas like differential equations and complex analysis. Mastering epsilon-delta proofs is a cornerstone of real analysis and crucial for developing rigorous mathematical reasoning.
#### Abstract Algebra: The Study of Structure
Abstract algebra moves beyond the familiar operations of arithmetic, focusing on the study of algebraic structures like groups, rings, and fields. These structures possess specific properties and relationships, allowing mathematicians to explore fundamental symmetries and patterns across diverse mathematical domains. Group theory, in particular, has far-reaching applications in physics, cryptography, and computer science.
#### Linear Algebra: The Mathematics of Transformations
Linear algebra deals with vector spaces, linear transformations, matrices, and systems of linear equations. It provides a powerful toolkit for solving problems in numerous fields, including computer graphics, machine learning, and quantum mechanics. Understanding eigenvalues and eigenvectors is particularly crucial for analyzing linear transformations and their effects.
Bridging the Gap: From Foundation to Application
The foundations of higher mathematics might seem abstract at first, but their power lies in their ability to unify and illuminate diverse mathematical areas. The rigorous thinking cultivated through studying these foundational concepts is transferable to other fields, improving problem-solving skills and enhancing critical thinking abilities. The abstract nature of the subject pushes you to think creatively and develop a deeper understanding of underlying principles.
Conclusion
Mastering the foundations of higher mathematics is a challenging yet immensely rewarding endeavor. By gaining a solid understanding of logic, set theory, real analysis, abstract algebra, and linear algebra, you lay the groundwork for exploring the vast and intricate landscape of advanced mathematical concepts. The skills and analytical prowess developed through this journey are invaluable, not only within mathematics itself but also across various scientific and technological disciplines. The journey is demanding, but the rewards are immeasurable.
FAQs
1. Is a strong background in calculus necessary before studying higher mathematics? While a solid understanding of calculus is helpful, especially for real analysis, many introductory courses in higher mathematics (especially abstract algebra and set theory) don't require extensive prior calculus knowledge.
2. What resources are available for learning the foundations of higher mathematics? Many excellent textbooks are available, catering to various levels of experience. Online courses (MOOCs) also provide accessible and interactive learning opportunities.
3. How long does it typically take to master the foundations of higher mathematics? The time required varies considerably depending on individual aptitude, prior mathematical background, and the intensity of study. However, expect a dedicated commitment spanning several years for a thorough understanding.
4. What career paths benefit from a strong understanding of higher mathematics? A strong foundation in higher mathematics opens doors to careers in academia, research, data science, finance, computer science, and various engineering disciplines.
5. Are there any prerequisites for studying the foundations of higher mathematics? A solid grasp of high school algebra and trigonometry is usually sufficient. A basic understanding of proof techniques is beneficial but not always strictly required in introductory courses.
| foundations of higher mathematics: Foundations of Higher Mathematics Daniel M. Fendel, Diane Resek, 1990 Foundations of Higher Mathematics: Exploration and Proof is the ideal text to bridge the crucial gap between the standard calculus sequence and upper division mathematics courses. The book takes a fresh approach to the subject: it asks students to explore mathematical principles on their own and challenges them to think like mathematicians. Two unique features-an exploration approach to mathematics and an intuitive and integrated presentation of logic based on predicate calculus-distinguish the book from the competition. Both features enable students to own the mathematics they're working on. As a result, your students develop a stronger motivation to tackle upper-level courses and gain a deeper understanding of concepts presented. |
| foundations of higher mathematics: Transition to Higher Mathematics Bob A. Dumas, John Edward McCarthy, 2007 This book is written for students who have taken calculus and want to learn what real mathematics is. |
| foundations of higher mathematics: Bridge to Higher Mathematics Sam Vandervelde, 2010 This engaging math textbook is designed to equip students who have completed a standard high school math curriculum with the tools and techniques that they will need to succeed in upper level math courses. Topics covered include logic and set theory, proof techniques, number theory, counting, induction, relations, functions, and cardinality. |
| foundations of higher mathematics: Homotopy Type Theory: Univalent Foundations of Mathematics , |
| foundations of higher mathematics: The Foundations of Mathematics Kenneth Kunen, 2009 Mathematical logic grew out of philosophical questions regarding the foundations of mathematics, but logic has now outgrown its philosophical roots, and has become an integral part of mathematics in general. This book is designed for students who plan to specialize in logic, as well as for those who are interested in the applications of logic to other areas of mathematics. Used as a text, it could form the basis of a beginning graduate-level course. There are three main chapters: Set Theory, Model Theory, and Recursion Theory. The Set Theory chapter describes the set-theoretic foundations of all of mathematics, based on the ZFC axioms. It also covers technical results about the Axiom of Choice, well-orderings, and the theory of uncountable cardinals. The Model Theory chapter discusses predicate logic and formal proofs, and covers the Completeness, Compactness, and Lowenheim-Skolem Theorems, elementary submodels, model completeness, and applications to algebra. This chapter also continues the foundational issues begun in the set theory chapter. Mathematics can now be viewed as formal proofs from ZFC. Also, model theory leads to models of set theory. This includes a discussion of absoluteness, and an analysis of models such as H( ) and R( ). The Recursion Theory chapter develops some basic facts about computable functions, and uses them to prove a number of results of foundational importance; in particular, Church's theorem on the undecidability of logical consequence, the incompleteness theorems of Godel, and Tarski's theorem on the non-definability of truth. |
| foundations of higher mathematics: Foundations of Analysis Joseph L. Taylor, 2012 Foundations of Analysis has two main goals. The first is to develop in students the mathematical maturity and sophistication they will need as they move through the upper division curriculum. The second is to present a rigorous development of both single and several variable calculus, beginning with a study of the properties of the real number system. The presentation is both thorough and concise, with simple, straightforward explanations. The exercises differ widely in level of abstraction and level of difficulty. They vary from the simple to the quite difficult and from the computational to the theoretical. Each section contains a number of examples designed to illustrate the material in the section and to teach students how to approach the exercises for that section. --Book cover. |
| foundations of higher mathematics: Foundations of Higher Mathematics Peter Fletcher, C. Wayne Patty, 1996 This text introduces students to basic techniques of writing proofs and acquaints them with some fundamental ideas. The authors assume that students using this text have already taken courses in which they developed the skill of using results and arguments that others have conceived. This text picks up where the others left off -- it develops the students' ability to think mathematically and to distinguish mathematical thinking from wishful thinking. |
| foundations of higher mathematics: A Bridge to Advanced Mathematics Dennis Sentilles, 2013-05-20 This helpful bridge book offers students the foundations they need to understand advanced mathematics. The two-part treatment provides basic tools and covers sets, relations, functions, mathematical proofs and reasoning, more. 1975 edition. |
| foundations of higher mathematics: Proofs and Fundamentals Ethan D. Bloch, 2013-12-01 The aim of this book is to help students write mathematics better. Throughout it are large exercise sets well-integrated with the text and varying appropriately from easy to hard. Basic issues are treated, and attention is given to small issues like not placing a mathematical symbol directly after a punctuation mark. And it provides many examples of what students should think and what they should write and how these two are often not the same. |
| foundations of higher mathematics: Real Analysis N. L. Carothers, 2000-08-15 A text for a first graduate course in real analysis for students in pure and applied mathematics, statistics, education, engineering, and economics. |
| foundations of higher mathematics: Introduction to the Foundations of Mathematics Raymond L. Wilder, 2013-09-26 Classic undergraduate text acquaints students with fundamental concepts and methods of mathematics. Topics include axiomatic method, set theory, infinite sets, groups, intuitionism, formal systems, mathematical logic, and much more. 1965 second edition. |
| foundations of higher mathematics: Higher Math for Beginners Y. B. Zeldovich, I. M. Yaglom, 1987 |
| foundations of higher mathematics: Mathematics for Machine Learning Marc Peter Deisenroth, A. Aldo Faisal, Cheng Soon Ong, 2020-04-23 Distills key concepts from linear algebra, geometry, matrices, calculus, optimization, probability and statistics that are used in machine learning. |
| foundations of higher mathematics: Logical Foundations of Mathematics and Computational Complexity Pavel Pudlák, 2013-04-22 The two main themes of this book, logic and complexity, are both essential for understanding the main problems about the foundations of mathematics. Logical Foundations of Mathematics and Computational Complexity covers a broad spectrum of results in logic and set theory that are relevant to the foundations, as well as the results in computational complexity and the interdisciplinary area of proof complexity. The author presents his ideas on how these areas are connected, what are the most fundamental problems and how they should be approached. In particular, he argues that complexity is as important for foundations as are the more traditional concepts of computability and provability. Emphasis is on explaining the essence of concepts and the ideas of proofs, rather than presenting precise formal statements and full proofs. Each section starts with concepts and results easily explained, and gradually proceeds to more difficult ones. The notes after each section present some formal definitions, theorems and proofs. Logical Foundations of Mathematics and Computational Complexity is aimed at graduate students of all fields of mathematics who are interested in logic, complexity and foundations. It will also be of interest for both physicists and philosophers who are curious to learn the basics of logic and complexity theory. |
| foundations of higher mathematics: Foundations of Computational Mathematics Ronald A. DeVore, Arieh Iserles, Endre Süli, 2001-05-17 Collection of papers by leading researchers in computational mathematics, suitable for graduate students and researchers. |
| foundations of higher mathematics: Introduction to the Mathematical and Statistical Foundations of Econometrics Herman J. Bierens, 2004-12-20 This book is intended for use in a rigorous introductory PhD level course in econometrics. |
| foundations of higher mathematics: Principia Mathematica Alfred North Whitehead, Bertrand Russell, 1927 The Principia Mathematica has long been recognised as one of the intellectual landmarks of the century. |
| foundations of higher mathematics: Foundations of Modern Analysis Avner Friedman, 1982-01-01 Measure and integration, metric spaces, the elements of functional analysis in Banach spaces, and spectral theory in Hilbert spaces — all in a single study. Only book of its kind. Unusual topics, detailed analyses. Problems. Excellent for first-year graduate students, almost any course on modern analysis. Preface. Bibliography. Index. |
| foundations of higher mathematics: A Transition to Advanced Mathematics William Johnston, Alex McAllister, 2009-07-27 A Transition to Advanced Mathematics: A Survey Course promotes the goals of a bridge'' course in mathematics, helping to lead students from courses in the calculus sequence (and other courses where they solve problems that involve mathematical calculations) to theoretical upper-level mathematics courses (where they will have to prove theorems and grapple with mathematical abstractions). The text simultaneously promotes the goals of a ``survey'' course, describing the intriguing questions and insights fundamental to many diverse areas of mathematics, including Logic, Abstract Algebra, Number Theory, Real Analysis, Statistics, Graph Theory, and Complex Analysis. The main objective is to bring about a deep change in the mathematical character of students -- how they think and their fundamental perspectives on the world of mathematics. This text promotes three major mathematical traits in a meaningful, transformative way: to develop an ability to communicate with precise language, to use mathematically sound reasoning, and to ask probing questions about mathematics. In short, we hope that working through A Transition to Advanced Mathematics encourages students to become mathematicians in the fullest sense of the word. A Transition to Advanced Mathematics has a number of distinctive features that enable this transformational experience. Embedded Questions and Reading Questions illustrate and explain fundamental concepts, allowing students to test their understanding of ideas independent of the exercise sets. The text has extensive, diverse Exercises Sets; with an average of 70 exercises at the end of section, as well as almost 3,000 distinct exercises. In addition, every chapter includes a section that explores an application of the theoretical ideas being studied. We have also interwoven embedded reflections on the history, culture, and philosophy of mathematics throughout the text. |
| foundations of higher mathematics: An Infinite Descent Into Pure Mathematics Clive Newstead, 2019-08 This introductory undergraduate-level textbook covers the knowledge and skills required to study pure mathematics at an advanced level. Emphasis is placed on communicating mathematical ideas precisely and effectively. A wide range of topic areas are covered. |
| foundations of higher mathematics: The Princeton Companion to Mathematics Timothy Gowers, June Barrow-Green, Imre Leader, 2010-07-18 The ultimate mathematics reference book This is a one-of-a-kind reference for anyone with a serious interest in mathematics. Edited by Timothy Gowers, a recipient of the Fields Medal, it presents nearly two hundred entries—written especially for this book by some of the world's leading mathematicians—that introduce basic mathematical tools and vocabulary; trace the development of modern mathematics; explain essential terms and concepts; examine core ideas in major areas of mathematics; describe the achievements of scores of famous mathematicians; explore the impact of mathematics on other disciplines such as biology, finance, and music—and much, much more. Unparalleled in its depth of coverage, The Princeton Companion to Mathematics surveys the most active and exciting branches of pure mathematics. Accessible in style, this is an indispensable resource for undergraduate and graduate students in mathematics as well as for researchers and scholars seeking to understand areas outside their specialties. Features nearly 200 entries, organized thematically and written by an international team of distinguished contributors Presents major ideas and branches of pure mathematics in a clear, accessible style Defines and explains important mathematical concepts, methods, theorems, and open problems Introduces the language of mathematics and the goals of mathematical research Covers number theory, algebra, analysis, geometry, logic, probability, and more Traces the history and development of modern mathematics Profiles more than ninety-five mathematicians who influenced those working today Explores the influence of mathematics on other disciplines Includes bibliographies, cross-references, and a comprehensive index Contributors include: Graham Allan, Noga Alon, George Andrews, Tom Archibald, Sir Michael Atiyah, David Aubin, Joan Bagaria, Keith Ball, June Barrow-Green, Alan Beardon, David D. Ben-Zvi, Vitaly Bergelson, Nicholas Bingham, Béla Bollobás, Henk Bos, Bodil Branner, Martin R. Bridson, John P. Burgess, Kevin Buzzard, Peter J. Cameron, Jean-Luc Chabert, Eugenia Cheng, Clifford C. Cocks, Alain Connes, Leo Corry, Wolfgang Coy, Tony Crilly, Serafina Cuomo, Mihalis Dafermos, Partha Dasgupta, Ingrid Daubechies, Joseph W. Dauben, John W. Dawson Jr., Francois de Gandt, Persi Diaconis, Jordan S. Ellenberg, Lawrence C. Evans, Florence Fasanelli, Anita Burdman Feferman, Solomon Feferman, Charles Fefferman, Della Fenster, José Ferreirós, David Fisher, Terry Gannon, A. Gardiner, Charles C. Gillispie, Oded Goldreich, Catherine Goldstein, Fernando Q. Gouvêa, Timothy Gowers, Andrew Granville, Ivor Grattan-Guinness, Jeremy Gray, Ben Green, Ian Grojnowski, Niccolò Guicciardini, Michael Harris, Ulf Hashagen, Nigel Higson, Andrew Hodges, F. E. A. Johnson, Mark Joshi, Kiran S. Kedlaya, Frank Kelly, Sergiu Klainerman, Jon Kleinberg, Israel Kleiner, Jacek Klinowski, Eberhard Knobloch, János Kollár, T. W. Körner, Michael Krivelevich, Peter D. Lax, Imre Leader, Jean-François Le Gall, W. B. R. Lickorish, Martin W. Liebeck, Jesper Lützen, Des MacHale, Alan L. Mackay, Shahn Majid, Lech Maligranda, David Marker, Jean Mawhin, Barry Mazur, Dusa McDuff, Colin McLarty, Bojan Mohar, Peter M. Neumann, Catherine Nolan, James Norris, Brian Osserman, Richard S. Palais, Marco Panza, Karen Hunger Parshall, Gabriel P. Paternain, Jeanne Peiffer, Carl Pomerance, Helmut Pulte, Bruce Reed, Michael C. Reed, Adrian Rice, Eleanor Robson, Igor Rodnianski, John Roe, Mark Ronan, Edward Sandifer, Tilman Sauer, Norbert Schappacher, Andrzej Schinzel, Erhard Scholz, Reinhard Siegmund-Schultze, Gordon Slade, David J. Spiegelhalter, Jacqueline Stedall, Arild Stubhaug, Madhu Sudan, Terence Tao, Jamie Tappenden, C. H. Taubes, Rüdiger Thiele, Burt Totaro, Lloyd N. Trefethen, Dirk van Dalen, Richard Weber, Dominic Welsh, Avi Wigderson, Herbert Wilf, David Wilkins, B. Yandell, Eric Zaslow, and Doron Zeilberger |
| foundations of higher mathematics: Foundations without Foundationalism Stewart Shapiro, 1991-09-19 The central contention of this book is that second-order logic has a central role to play in laying the foundations of mathematics. In order to develop the argument fully, the author presents a detailed development of higher-order logic, including a comprehensive discussion of its semantics. Professor Shapiro demonstrates the prevalence of second-order notions in mathematics is practised, and also the extent to which mathematical concepts can be formulated in second-order languages . He shows how first-order languages are insufficient to codify many concepts in contemporary mathematics, and thus that higher-order logic is needed to fully reflect current mathematics. Throughout, the emphasis is on discussing the philosophical and historical issues associated with this subject, and the implications that they have for foundational studies. For the most part, the author assumes little more than a familiarity with logic as might be gained from a beginning graduate course which includes the incompleteness of arithmetic and the Lowenheim-Skolem theorems. All those concerned with the foundations of mathematics will find this a thought-provoking discussion of some of the central issues in this subject. |
| foundations of higher mathematics: Reflections on the Foundations of Mathematics Stefania Centrone, Deborah Kant, Deniz Sarikaya, 2019-11-11 This edited work presents contemporary mathematical practice in the foundational mathematical theories, in particular set theory and the univalent foundations. It shares the work of significant scholars across the disciplines of mathematics, philosophy and computer science. Readers will discover systematic thought on criteria for a suitable foundation in mathematics and philosophical reflections around the mathematical perspectives. The volume is divided into three sections, the first two of which focus on the two most prominent candidate theories for a foundation of mathematics. Readers may trace current research in set theory, which has widely been assumed to serve as a framework for foundational issues, as well as new material elaborating on the univalent foundations, considering an approach based on homotopy type theory (HoTT). The third section then builds on this and is centred on philosophical questions connected to the foundations of mathematics. Here, the authors contribute to discussions on foundational criteria with more general thoughts on the foundations of mathematics which are not connected to particular theories. This book shares the work of some of the most important scholars in the fields of set theory (S. Friedman), non-classical logic (G. Priest) and the philosophy of mathematics (P. Maddy). The reader will become aware of the advantages of each theory and objections to it as a foundation, following the latest and best work across the disciplines and it is therefore a valuable read for anyone working on the foundations of mathematics or in the philosophy of mathematics. |
| foundations of higher mathematics: 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. |
| foundations of higher mathematics: Cognitive Foundations for Improving Mathematical Learning David C. Geary, Daniel B. Berch, Kathleen Mann Koepke, 2019-01-03 The fifth volume in the Mathematical Cognition and Learning series focuses on informal learning environments and other parental influences on numerical cognitive development and formal instructional interventions for improving mathematics learning and performance. The chapters cover the use of numerical play and games for improving foundational number knowledge as well as school math performance, the link between early math abilities and the approximate number system, and how families can help improve the early development of math skills. The book goes on to examine learning trajectories in early mathematics, the role of mathematical language in acquiring numeracy skills, evidence-based assessments of early math skills, approaches for intensifying early mathematics interventions, the use of analogies in mathematics instruction, schema-based diagrams for teaching ratios and proportions, the role of cognitive processes in treating mathematical learning difficulties, and addresses issues associated with intervention fadeout. - Identifies the relative influence of school and family on math learning - Discusses the efficacy of numerical play for improvement in math - Features learning trajectories in math - Examines the role of math language in numeracy skills - Includes assessments of math skills - Explores the role of cognition in treating math-based learning difficulties |
| foundations of higher mathematics: Foundations of Modern Probability Olav Kallenberg, 2002-01-08 The first edition of this single volume on the theory of probability has become a highly-praised standard reference for many areas of probability theory. Chapters from the first edition have been revised and corrected, and this edition contains four new chapters. New material covered includes multivariate and ratio ergodic theorems, shift coupling, Palm distributions, Harris recurrence, invariant measures, and strong and weak ergodicity. |
| foundations of higher mathematics: Foundations of Mathematical Optimization Diethard Ernst Pallaschke, S. Rolewicz, 2013-03-14 Many books on optimization consider only finite dimensional spaces. This volume is unique in its emphasis: the first three chapters develop optimization in spaces without linear structure, and the analog of convex analysis is constructed for this case. Many new results have been proved specially for this publication. In the following chapters optimization in infinite topological and normed vector spaces is considered. The novelty consists in using the drop property for weak well-posedness of linear problems in Banach spaces and in a unified approach (by means of the Dolecki approximation) to necessary conditions of optimality. The method of reduction of constraints for sufficient conditions of optimality is presented. The book contains an introduction to non-differentiable and vector optimization. Audience: This volume will be of interest to mathematicians, engineers, and economists working in mathematical optimization. |
| foundations of higher mathematics: A Course of Higher Mathematics Vladimir Ivanovich Smirnov, 1964 |
| foundations of higher mathematics: Foundations of Real and Abstract Analysis Douglas S. Bridges, 2014-01-15 |
| foundations of higher mathematics: Fundamentals of Mathematics Denny Burzynski, Wade Ellis, 2008 Fundamentals of Mathematics is a work text that covers the traditional study in a modern prealgebra course, as well as the topics of estimation, elementary analytic geometry, and introductory algebra. It is intended for students who: have had previous courses in prealgebra wish to meet the prerequisites of higher level courses such as elementary algebra need to review fundamental mathematical concenpts and techniques This text will help the student devlop the insight and intuition necessary to master arithmetic techniques and manipulative skills. It was written with the following main objectives: to provide the student with an understandable and usable source of information to provide the student with the maximum oppurtinity to see that arithmetic concepts and techniques are logically based to instill in the student the understanding and intuitive skills necessary to know how and when to use particular arithmetic concepts in subsequent material cources and nonclassroom situations to give the students the ability to correctly interpret arithmetically obtained results We have tried to meet these objects by presenting material dynamically much the way an instructure might present the material visually in a classroom. (See the development of the concept of addition and subtraction of fractions in section 5.3 for examples) Intuition and understanding are some of the keys to creative thinking, we belive that the material presented in this text will help students realize that mathematics is a creative subject. |
| foundations of higher mathematics: College Accounting John Ellis Price, M. David Haddock, Horace R. Brock, 2007 |
| foundations of higher mathematics: Mathematical Reasoning Theodore A. Sundstrom, 2007 Focusing on the formal development of mathematics, this book shows readers how to read, understand, write, and construct mathematical proofs.Uses elementary number theory and congruence arithmetic throughout. Focuses on writing in mathematics. Reviews prior mathematical work with “Preview Activities” at the start of each section. Includes “Activities” throughout that relate to the material contained in each section. Focuses on Congruence Notation and Elementary Number Theorythroughout.For professionals in the sciences or engineering who need to brush up on their advanced mathematics skills. Mathematical Reasoning: Writing and Proof, 2/E Theodore Sundstrom |
| foundations of higher mathematics: A Foundation Course in Mathematics Ajit Kumar, S. Kumaresan, B. K. Sarma, 2018-04-30 Written in a conversational style to impart critical and analytical thinking which will be beneficial for students of any discipline. It also gives emphasis on problem solving and proof writing skills, key aspects of learning mathematics. |
| foundations of higher mathematics: Foundations of Computation Carol Critchlow, David Eck, 2011 Foundations of Computation is a free textbook for a one-semester course in theoretical computer science. It has been used for several years in a course at Hobart and William Smith Colleges. The course has no prerequisites other than introductory computer programming. The first half of the course covers material on logic, sets, and functions that would often be taught in a course in discrete mathematics. The second part covers material on automata, formal languages and grammar that would ordinarily be encountered in an upper level course in theoretical computer science. |
| foundations of higher mathematics: Book of Proof Richard H. Hammack, 2016-01-01 This book is an introduction to the language and standard proof methods of mathematics. It is a bridge from the computational courses (such as calculus or differential equations) that students typically encounter in their first year of college to a more abstract outlook. It lays a foundation for more theoretical courses such as topology, analysis and abstract algebra. Although it may be more meaningful to the student who has had some calculus, there is really no prerequisite other than a measure of mathematical maturity. |
| foundations of higher mathematics: Introduction to Mathematical Thinking Keith J. Devlin, 2012 Mathematical thinking is not the same as 'doing math'--unless you are a professional mathematician. For most people, 'doing math' means the application of procedures and symbolic manipulations. Mathematical thinking, in contrast, is what the name reflects, a way of thinking about things in the world that humans have developed over three thousand years. It does not have to be about mathematics at all, which means that many people can benefit from learning this powerful way of thinking, not just mathematicians and scientists.--Back cover. |
| foundations of higher mathematics: Basic Mathematics Serge Lang, 1988-01 |
| foundations of higher mathematics: Foundations of Mathematical Real Analysis: Computer Science Mathematical Analysis Chidume O. C, 2019-08-29 This book is intended as a serious introduction to the studyof mathematical analysis. In contrast to calculus, mathematical analysis does not involve formula manipulation, memorizing integrals or applications to other fields of science. No.It involves geometric intuition and proofs of theorems. It ispure mathematics! Given the mathematical preparation andinterest of our intended audience which, apart from mathematics majors, includes students of statistics, computer science, physics, students of mathematics education and students of engineering, we have not given the axiomatic development of the real number system. However, we assumethat the reader is familiar with sets and functions. This bookis divided into two parts. Part I covers elements of mathematical analysis which include: the real number system, bounded subsets of real numbers, sequences of real numbers, monotone sequences, Bolzano-Weierstrass theorem, Cauchysequences and completeness of R, continuity, intermediatevalue theorem, continuous maps on [a, b], uniform continuity, closed sets, compact sets, differentiability, series of nonnegative real numbers, alternating series, absolute and conditional convergence; and re-arrangement of series. The contents of Part I are adequate for a semester course in mathematical analysis at the 200 level. Part II covers Riemannintegrals. In particular, the Riemann integral, basic properties of Riemann integral, pointwise convergence of sequencesof functions, uniform convergence of sequences of functions, series of real-valued functions: term by term differentiationand integration; power series: uniform convergence of powerseries; uniform convergence at end points; and equi-continuity are covered. Part II covers the standard syllabus for asemester mathematical analysis course at the 300 level. Thetopics covered in this book provide a reasonable preparationfor any serious study of higher mathematics. But for one toreally benefit from the book, one must spend a great deal ofixtime on it, studying the contents very carefully and attempting all the exercises, especially the miscellaneous exercises atthe end of the book. These exercises constitute an importantintegral part of the book.Each chapter begins with clear statements of the most important theorems of the chapter. The proofs of these theoremsgenerally contain fundamental ideas of mathematical analysis. Students are therefore encouraged to study them verycarefully and to discover these id |
| foundations of higher mathematics: Fundamentals of Mathematics , 1974 |
| foundations of higher mathematics: Foundations of Infinitesimal Calculus H. Jerome Keisler, 1976-01-01 |
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