Proof, Logic, and Conjecture

The Mathematician's Toolbox
Author: Robert S. Wolf
Publisher: St. Martin's Press
ISBN: 9780716730507
Category: Mathematics
Page: 421
View: 9803

Continue Reading →

This text is designed to teach students how to read and write proofs in mathematics and to acquaint them with how mathematicians investigate problems and formulate conjecture.

A Tour Through Mathematical Logic


Author: Robert S. Wolf
Publisher: MAA
ISBN: 9780883850367
Category: Mathematics
Page: 397
View: 9405

Continue Reading →

A guide that any interested reader with some post-calculus experience in mathematics can read, enjoy, and learn from.

A Survey of Modern Algebra


Author: Garrett Birkhoff,Saunders Mac Lane
Publisher: CRC Press
ISBN: 1351991736
Category: Mathematics
Page: 512
View: 4143

Continue Reading →

This classic, written by two young instructors who became giants in their field, has shaped the understanding of modern algebra for generations of mathematicians and remains a valuable reference and text for self study and college courses.

Mathematical Reasoning

Patterns, Problems, Conjectures, and Proofs
Author: Raymond Nickerson
Publisher: Taylor & Francis
ISBN: 1136945393
Category: Psychology
Page: 595
View: 8143

Continue Reading →

The development of mathematical competence -- both by humans as a species over millennia and by individuals over their lifetimes -- is a fascinating aspect of human cognition. This book explores a vast range of psychological questions related to mathematical cognition, and provides fascinating insights for researchers and students of cognition and instructors of mathematics.

How to Think Like a Mathematician

A Companion to Undergraduate Mathematics
Author: Kevin Houston
Publisher: Cambridge University Press
ISBN: 9781139477055
Category: Mathematics
Page: N.A
View: 7437

Continue Reading →

Looking for a head start in your undergraduate degree in mathematics? Maybe you've already started your degree and feel bewildered by the subject you previously loved? Don't panic! This friendly companion will ease your transition to real mathematical thinking. Working through the book you will develop an arsenal of techniques to help you unlock the meaning of definitions, theorems and proofs, solve problems, and write mathematics effectively. All the major methods of proof - direct method, cases, induction, contradiction and contrapositive - are featured. Concrete examples are used throughout, and you'll get plenty of practice on topics common to many courses such as divisors, Euclidean algorithms, modular arithmetic, equivalence relations, and injectivity and surjectivity of functions. The material has been tested by real students over many years so all the essentials are covered. With over 300 exercises to help you test your progress, you'll soon learn how to think like a mathematician.

Discrete Mathematics

Proofs, Structures and Applications, Third Edition
Author: Rowan Garnier,John Taylor
Publisher: Taylor & Francis
ISBN: 1439812810
Category: Mathematics
Page: 843
View: 2333

Continue Reading →

Taking an approach to the subject that is suitable for a broad readership, Discrete Mathematics: Proofs, Structures, and Applications, Third Edition provides a rigorous yet accessible exposition of discrete mathematics, including the core mathematical foundation of computer science. The approach is comprehensive yet maintains an easy-to-follow progression from the basic mathematical ideas to the more sophisticated concepts examined later in the book. This edition preserves the philosophy of its predecessors while updating and revising some of the content. New to the Third Edition In the expanded first chapter, the text includes a new section on the formal proof of the validity of arguments in propositional logic before moving on to predicate logic. This edition also contains a new chapter on elementary number theory and congruences. This chapter explores groups that arise in modular arithmetic and RSA encryption, a widely used public key encryption scheme that enables practical and secure means of encrypting data. This third edition also offers a detailed solutions manual for qualifying instructors. Exploring the relationship between mathematics and computer science, this text continues to provide a secure grounding in the theory of discrete mathematics and to augment the theoretical foundation with salient applications. It is designed to help readers develop the rigorous logical thinking required to adapt to the demands of the ever-evolving discipline of computer science.

A Tour Through Mathematical Logic


Author: Robert S. Wolf
Publisher: MAA
ISBN: 9780883850367
Category: Mathematics
Page: 397
View: 1470

Continue Reading →

A guide that any interested reader with some post-calculus experience in mathematics can read, enjoy, and learn from.

Introduction to Advanced Mathematics: A Guide to Understanding Proofs


Author: Connie M. Campbell
Publisher: Cengage Learning
ISBN: 0547165382
Category: Mathematics
Page: 144
View: 9127

Continue Reading →

This text offers a crucial primer on proofs and the language of mathematics. Brief and to the point, it lays out the fundamental ideas of abstract mathematics and proof techniques that students will need to master for other math courses. Campbell presents these concepts in plain English, with a focus on basic terminology and a conversational tone that draws natural parallels between the language of mathematics and the language students communicate in every day. The discussion highlights how symbols and expressions are the building blocks of statements and arguments, the meanings they convey, and why they are meaningful to mathematicians. In-class activities provide opportunities to practice mathematical reasoning in a live setting, and an ample number of homework exercises are included for self-study. This text is appropriate for a course in Foundations of Advanced Mathematics taken by students who've had a semester of calculus, and is designed to be accessible to students with a wide range of mathematical proficiency. It can also be used as a self-study reference, or as a supplement in other math courses where additional proofs practice is needed. Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version.

An Invitation to Real Analysis


Author: Luis F. Moreno
Publisher: The Mathematical Association of America
ISBN: 1939512050
Category: Mathematics
Page: 680
View: 1590

Continue Reading →

An Invitation to Real Analysis is written both as a stepping stone to higher calculus and analysis courses, and as foundation for deeper reasoning in applied mathematics. This book also provides a broader foundation in real analysis than is typical for future teachers of secondary mathematics. In connection with this, within the chapters, students are pointed to numerous articles from The College Mathematics Journal and The American Mathematical Monthly. These articles are inviting in their level of exposition and their wide-ranging content. Axioms are presented with an emphasis on the distinguishing characteristics that new ones bring, culminating with the axioms that define the reals. Set theory is another theme found in this book, beginning with what students are familiar with from basic calculus. This theme runs underneath the rigorous development of functions, sequences, and series, and then ends with a chapter on transfinite cardinal numbers and with chapters on basic point-set topology. Differentiation and integration are developed with the standard level of rigor, but always with the goal of forming a firm foundation for the student who desires to pursue deeper study. A historical theme interweaves throughout the book, with many quotes and accounts of interest to all readers. Over 600 exercises and dozens of figures help the learning process. Several topics (continued fractions, for example), are included in the appendices as enrichment material. An annotated bibliography is included.

Numbers and Proofs


Author: Reg Allenby
Publisher: Elsevier
ISBN: 0080928773
Category: Mathematics
Page: 288
View: 2295

Continue Reading →

'Numbers and Proofs' presents a gentle introduction to the notion of proof to give the reader an understanding of how to decipher others' proofs as well as construct their own. Useful methods of proof are illustrated in the context of studying problems concerning mainly numbers (real, rational, complex and integers). An indispensable guide to all students of mathematics. Each proof is preceded by a discussion which is intended to show the reader the kind of thoughts they might have before any attempt proof is made. Established proofs which the student is in a better position to follow then follow. Presented in the author's entertaining and informal style, and written to reflect the changing profile of students entering universities, this book will prove essential reading for all seeking an introduction to the notion of proof as well as giving a definitive guide to the more common forms. Stressing the importance of backing up "truths" found through experimentation, with logically sound and watertight arguments, it provides an ideal bridge to more complex undergraduate maths.

The Proof is in the Pudding

The Changing Nature of Mathematical Proof
Author: Steven G. Krantz
Publisher: Springer Science & Business Media
ISBN: 9780387487441
Category: Mathematics
Page: 264
View: 4658

Continue Reading →

This text explores the many transformations that the mathematical proof has undergone from its inception to its versatile, present-day use, considering the advent of high-speed computing machines. Though there are many truths to be discovered in this book, by the end it is clear that there is no formalized approach or standard method of discovery to date. Most of the proofs are discussed in detail with figures and equations accompanying them, allowing both the professional mathematician and those less familiar with mathematics to derive the same joy from reading this book.

Proofs and Refutations

The Logic of Mathematical Discovery
Author: Imre Lakatos
Publisher: Cambridge University Press
ISBN: 1316425339
Category: Science
Page: N.A
View: 7069

Continue Reading →

Imre Lakatos's Proofs and Refutations is an enduring classic, which has never lost its relevance. Taking the form of a dialogue between a teacher and some students, the book considers various solutions to mathematical problems and, in the process, raises important questions about the nature of mathematical discovery and methodology. Lakatos shows that mathematics grows through a process of improvement by attempts at proofs and critiques of these attempts, and his work continues to inspire mathematicians and philosophers aspiring to develop a philosophy of mathematics that accounts for both the static and the dynamic complexity of mathematical practice. With a specially commissioned Preface written by Paolo Mancosu, this book has been revived for a new generation of readers.

The History of Mathematical Proof in Ancient Traditions


Author: Karine Chemla
Publisher: Cambridge University Press
ISBN: 1139510584
Category: Philosophy
Page: N.A
View: 7621

Continue Reading →

This radical, profoundly scholarly book explores the purposes and nature of proof in a range of historical settings. It overturns the view that the first mathematical proofs were in Greek geometry and rested on the logical insights of Aristotle by showing how much of that view is an artefact of nineteenth-century historical scholarship. It documents the existence of proofs in ancient mathematical writings about numbers and shows that practitioners of mathematics in Mesopotamian, Chinese and Indian cultures knew how to prove the correctness of algorithms, which are much more prominent outside the limited range of surviving classical Greek texts that historians have taken as the paradigm of ancient mathematics. It opens the way to providing the first comprehensive, textually based history of proof.

Conjecture and Proof


Author: Miklós Laczkovich
Publisher: Cambridge University Press
ISBN: 9780883857229
Category: Mathematics
Page: 118
View: 5903

Continue Reading →

How to prove interesting and deep mathematical results from first principles, with exercises.

Prime Obsession

Bernhard Riemann and the Greatest Unsolved Problem in Mathematics
Author: John Derbyshire
Publisher: Joseph Henry Press
ISBN: 0309141257
Category: Science
Page: 446
View: 1217

Continue Reading →

In August 1859 Bernhard Riemann, a little-known 32-year old mathematician, presented a paper to the Berlin Academy titled: "On the Number of Prime Numbers Less Than a Given Quantity." In the middle of that paper, Riemann made an incidental remark â€" a guess, a hypothesis. What he tossed out to the assembled mathematicians that day has proven to be almost cruelly compelling to countless scholars in the ensuing years. Today, after 150 years of careful research and exhaustive study, the question remains. Is the hypothesis true or false? Riemann's basic inquiry, the primary topic of his paper, concerned a straightforward but nevertheless important matter of arithmetic â€" defining a precise formula to track and identify the occurrence of prime numbers. But it is that incidental remark â€" the Riemann Hypothesis â€" that is the truly astonishing legacy of his 1859 paper. Because Riemann was able to see beyond the pattern of the primes to discern traces of something mysterious and mathematically elegant shrouded in the shadows â€" subtle variations in the distribution of those prime numbers. Brilliant for its clarity, astounding for its potential consequences, the Hypothesis took on enormous importance in mathematics. Indeed, the successful solution to this puzzle would herald a revolution in prime number theory. Proving or disproving it became the greatest challenge of the age. It has become clear that the Riemann Hypothesis, whose resolution seems to hang tantalizingly just beyond our grasp, holds the key to a variety of scientific and mathematical investigations. The making and breaking of modern codes, which depend on the properties of the prime numbers, have roots in the Hypothesis. In a series of extraordinary developments during the 1970s, it emerged that even the physics of the atomic nucleus is connected in ways not yet fully understood to this strange conundrum. Hunting down the solution to the Riemann Hypothesis has become an obsession for many â€" the veritable "great white whale" of mathematical research. Yet despite determined efforts by generations of mathematicians, the Riemann Hypothesis defies resolution. Alternating passages of extraordinarily lucid mathematical exposition with chapters of elegantly composed biography and history, Prime Obsession is a fascinating and fluent account of an epic mathematical mystery that continues to challenge and excite the world. Posited a century and a half ago, the Riemann Hypothesis is an intellectual feast for the cognoscenti and the curious alike. Not just a story of numbers and calculations, Prime Obsession is the engrossing tale of a relentless hunt for an elusive proof â€" and those who have been consumed by it.

Philosophy of Mathematics

Selected Writings
Author: Charles S. Peirce
Publisher: Indiana University Press
ISBN: 0253004691
Category: Philosophy
Page: 336
View: 1746

Continue Reading →

The philosophy of mathematics plays a vital role in the mature philosophy of Charles S. Peirce. Peirce received rigorous mathematical training from his father and his philosophy carries on in decidedly mathematical and symbolic veins. For Peirce, math was a philosophical tool and many of his most productive ideas rest firmly on the foundation of mathematical principles. This volume collects Peirce's most important writings on the subject, many appearing in print for the first time. Peirce's determination to understand matter, the cosmos, and "the grand design" of the universe remain relevant for contemporary students of science, technology, and symbolic logic.

What is Mathematics?

An Elementary Approach to Ideas and Methods
Author: Richard Courant,Herbert Robbins,Ian Stewart
Publisher: Oxford University Press, USA
ISBN: 9780195105193
Category: Mathematics
Page: 566
View: 8164

Continue Reading →

A discussion of fundamental mathematical principles from algebra to elementary calculus designed to promote constructive mathematical reasoning.