• Fiction

The Foundations of Mathematics


Author: Ian Stewart,David Orme Tall
Publisher: Oxford University Press on Demand
ISBN: 9780198531654
Category: Fiction
Page: 263
View: 1148
"There are many textbooks available for a so-called transition course from calculus to abstract mathematics. I have taught this course several times and always find it problematic. The Foundations of Mathematics (Stewart and Tall) is a horse of a different color. The writing is excellent and there is actually some useful mathematics. I definitely like this book."--The Bulletin of Mathematics Books

    • Mathematics

Introduction to the Foundations of Mathematics

Second Edition
Author: Raymond L. Wilder
Publisher: Courier Corporation
ISBN: 0486276201
Category: Mathematics
Page: 352
View: 869
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.

    • Mathematics

The Foundations of Mathematics


Author: Kenneth Kunen
Publisher: N.A
ISBN: 9781904987147
Category: Mathematics
Page: 251
View: 3811
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.

    • Philosophy

Foundations of Mathematics and other Logical Essays


Author: Frank Plumpton Ramsey
Publisher: Routledge
ISBN: 1134528035
Category: Philosophy
Page: 312
View: 4058
First published in 2000. Routledge is an imprint of Taylor & Francis, an informa company.

    • Mathematics

The Foundations of Mathematics in the Theory of Sets


Author: John P. Mayberry
Publisher: Cambridge University Press
ISBN: 9780521770347
Category: Mathematics
Page: 424
View: 9815
This 2001 book will appeal to mathematicians and philosophers interested in the foundations of mathematics.

    • Mathematics

The Foundations of Mathematics


Author: Ian Stewart,David Tall
Publisher: Oxford University Press, USA
ISBN: 019870643X
Category: Mathematics
Page: 432
View: 7285
The transition from school mathematics to university mathematics is seldom straightforward. Students are faced with a disconnect between the algorithmic and informal attitude to mathematics at school, versus a new emphasis on proof, based on logic, and a more abstract development of general concepts, based on set theory. The authors have many years' experience of the potential difficulties involved, through teaching first-year undergraduates and researching the ways in which students and mathematicians think. The book explains the motivation behind abstract foundational material based on students' experiences of school mathematics, and explicitly suggests ways students can make sense of formal ideas. This second edition takes a significant step forward by not only making the transition from intuitive to formal methods, but also by reversing the process- using structure theorems to prove that formal systems have visual and symbolic interpretations that enhance mathematical thinking. This is exemplified by a new chapter on the theory of groups. While the first edition extended counting to infinite cardinal numbers, the second also extends the real numbers rigorously to larger ordered fields. This links intuitive ideas in calculus to the formal epsilon-delta methods of analysis. The approach here is not the conventional one of 'nonstandard analysis', but a simpler, graphically based treatment which makes the notion of an infinitesimal natural and straightforward. This allows a further vision of the wider world of mathematical thinking in which formal definitions and proof lead to amazing new ways of defining, proving, visualising and symbolising mathematics beyond previous expectations.


    • Mathematics

Kurt Gödel and the Foundations of Mathematics

Horizons of Truth
Author: Matthias Baaz,Christos H. Papadimitriou,Hilary W. Putnam,Dana S. Scott,Charles L. Harper, Jr
Publisher: Cambridge University Press
ISBN: 1139498436
Category: Mathematics
Page: N.A
View: 5321
This volume commemorates the life, work and foundational views of Kurt Gödel (1906–78), most famous for his hallmark works on the completeness of first-order logic, the incompleteness of number theory, and the consistency - with the other widely accepted axioms of set theory - of the axiom of choice and of the generalized continuum hypothesis. It explores current research, advances and ideas for future directions not only in the foundations of mathematics and logic, but also in the fields of computer science, artificial intelligence, physics, cosmology, philosophy, theology and the history of science. The discussion is supplemented by personal reflections from several scholars who knew Gödel personally, providing some interesting insights into his life. By putting his ideas and life's work into the context of current thinking and perceptions, this book will extend the impact of Gödel's fundamental work in mathematics, logic, philosophy and other disciplines for future generations of researchers.

    • Mathematics

The Foundations of Mathematics


Author: Thomas Q. Sibley
Publisher: John Wiley & Sons
ISBN: 0470085010
Category: Mathematics
Page: 392
View: 6591
Finally there's an easy-to-follow book that will help readers succeed in the art of proving theorems. Sibley not only conveys the spirit of mathematics but also uncovers the skills required to succeed. Key definitions are introduced while readers are encouraged to develop an intuition about these concepts and practice using them in problems. With this approach, they'll gain a strong understanding of the mathematical language as they discover how to apply it in order to find proofs.

    • Mathematics

Wittgenstein's Lectures on the Foundations of Mathematics, Cambridge, 1939


Author: Ludwig Wittgenstein,R. G. Bosanquet,Cora Diamond
Publisher: University of Chicago Press
ISBN: 9780226904269
Category: Mathematics
Page: 300
View: 2765
For several terms at Cambridge in 1939, Ludwig Wittgenstein lectured on the philosophical foundations of mathematics. A lecture class taught by Wittgenstein, however, hardly resembled a lecture. He sat on a chair in the middle of the room, with some of the class sitting in chairs, some on the floor. He never used notes. He paused frequently, sometimes for several minutes, while he puzzled out a problem. He often asked his listeners questions and reacted to their replies. Many meetings were largely conversation. These lectures were attended by, among others, D. A. T. Gasking, J. N. Findlay, Stephen Toulmin, Alan Turing, G. H. von Wright, R. G. Bosanquet, Norman Malcolm, Rush Rhees, and Yorick Smythies. Notes taken by these last four are the basis for the thirty-one lectures in this book. The lectures covered such topics as the nature of mathematics, the distinctions between mathematical and everyday languages, the truth of mathematical propositions, consistency and contradiction in formal systems, the logicism of Frege and Russell, Platonism, identity, negation, and necessary truth. The mathematical examples used are nearly always elementary.

    • Mathematics

Wittgenstein, Finitism, and the Foundations of Mathematics


Author: Mathieu Marion
Publisher: Oxford University Press
ISBN: 9780198235163
Category: Mathematics
Page: 260
View: 9505
Mathieu Marion offers a careful, historically informed study of Wittgenstein's philosophy of mathematics. This area of his work has frequently been undervalued by Wittgenstein specialists and philosophers of mathematics alike; but the surprising fact that he wrote more on this subject than any other indicates its centrality in his thought. Marion traces the development of Wittgenstein's thinking from the 1920s through to the 1950s, in the context of themathematical and philosophical work of the times, to make coherent sense of ideas that have too often been misunderstood because they have been presented in a disjointed and incomplete way. He shows that study of Wittgenstein's writings on mathematics is essential to a proper understanding of his philosophy,and also that it can do much to illuminate current debates about the foundations of mathematics.

    • Mathematics

Harvey Friedman's Research on the Foundations of Mathematics


Author: L.A. Harrington,M.D. Morley,A. Šcedrov,S.G. Simpson
Publisher: Elsevier
ISBN: 9780080960401
Category: Mathematics
Page: 407
View: 5626
This volume discusses various aspects of Harvey Friedman's research in the foundations of mathematics over the past fifteen years. It should appeal to a wide audience of mathematicians, computer scientists, and mathematically oriented philosophers.

    • Mathematics

The Logical Foundations of Mathematics

Foundations and Philosophy of Science and Technology Series
Author: William S. Hatcher
Publisher: Elsevier
ISBN: 1483189635
Category: Mathematics
Page: 330
View: 7548
The Logical Foundations of Mathematics offers a study of the foundations of mathematics, stressing comparisons between and critical analyses of the major non-constructive foundational systems. The position of constructivism within the spectrum of foundational philosophies is discussed, along with the exact relationship between topos theory and set theory. Comprised of eight chapters, this book begins with an introduction to first-order logic. In particular, two complete systems of axioms and rules for the first-order predicate calculus are given, one for efficiency in proving metatheorems, and the other, in a "natural deduction" style, for presenting detailed formal proofs. A somewhat novel feature of this framework is a full semantic and syntactic treatment of variable-binding term operators as primitive symbols of logic. Subsequent chapters focus on the origin of modern foundational studies; Gottlob Frege's formal system intended to serve as a foundation for mathematics and its paradoxes; the theory of types; and the Zermelo-Fraenkel set theory. David Hilbert's program and Kurt Gödel's incompleteness theorems are also examined, along with the foundational systems of W. V. Quine and the relevance of categorical algebra for foundations. This monograph will be of interest to students, teachers, practitioners, and researchers in mathematics.

    • Science

Foundations of Mathematical Genetics


Author: Anthony W. F. Edwards
Publisher: Cambridge University Press
ISBN: 9780521775441
Category: Science
Page: 121
View: 9169
Foundations of Mathematical Genetics is a definitive account of the origins of modern mathematical population genetics, a topic that has been all too often neglected by other textbooks. In this new edition, Dr Edwards extends his classic work with a new chapter on the fundamental theorem of natural selection. Advanced students of mathematical genetics and those interested in the history of the subject will find it a clear exposition of the mathematical underpinnings of population genetics.

    • Mathematics

Handbook of Mathematical Logic


Author: J. Barwise
Publisher: Elsevier
ISBN: 9780080933641
Category: Mathematics
Page: 1164
View: 9971
The handbook is divided into four parts: model theory, set theory, recursion theory and proof theory. Each of the four parts begins with a short guide to the chapters that follow. Each chapter is written for non-specialists in the field in question. Mathematicians will find that this book provides them with a unique opportunity to apprise themselves of developments in areas other than their own.


    • Mathematics

Practical Foundations of Mathematics


Author: Paul Taylor
Publisher: Cambridge University Press
ISBN: 9780521631075
Category: Mathematics
Page: 572
View: 1472
Practical Foundations collects the methods of construction of the objects of twentieth-century mathematics. Although it is mainly concerned with a framework essentially equivalent to intuitionistic Zermelo-Fraenkel logic, the book looks forward to more subtle bases in categorical type theory and the machine representation of mathematics. Each idea is illustrated by wide-ranging examples, and followed critically along its natural path, transcending disciplinary boundaries between universal algebra, type theory, category theory, set theory, sheaf theory, topology and programming. Students and teachers of computing, mathematics and philosophy will find this book both readable and of lasting value as a reference work.

    • Mathematics

Wittgenstein on the Foundations of Mathematics


Author: Crispin Wright
Publisher: N.A
ISBN: 9780751202953
Category: Mathematics
Page: 481
View: 692
In this detailed account, Crispin Wright offers a systematic account of Wittgenstein's later philosophy of mathematics and establishes its links with his later philosophy of language. In line with this, he examines Wittgenstein's Remarks on the Foundations of Mathematics.

    • Mathematics

Logical Foundations of Mathematics and Computational Complexity

A Gentle Introduction
Author: Pavel Pudlák
Publisher: Springer Science & Business Media
ISBN: 3319001191
Category: Mathematics
Page: 695
View: 8522
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.

    • Mathematics

Essays on the Foundations of Mathematics by Moritz Pasch


Author: Stephen Pollard
Publisher: Springer Science & Business Media
ISBN: 9789048194162
Category: Mathematics
Page: 248
View: 8177
Moritz Pasch (1843-1930) is justly celebrated as a key figure in the history of axiomatic geometry. Less well known are his contributions to other areas of foundational research. This volume features English translations of 14 papers Pasch published in the decade 1917-1926. In them, Pasch argues that geometry and, more surprisingly, number theory are branches of empirical science; he provides axioms for the combinatorial reasoning essential to Hilbert’s program of consistency proofs; he explores "implicit definition" (a generalization of definition by abstraction) and indicates how this technique yields an "empiricist" reconstruction of set theory; he argues that we cannot fully understand the logical structure of mathematics without clearly distinguishing between decidable and undecidable properties; he offers a rare glimpse into the mind of a master of axiomatics, surveying in detail the thought experiments he employed as he struggled to identify fundamental mathematical principles; and much more. This volume will: Give English speakers access to an important body of work from a turbulent and pivotal period in the history of mathematics, help us look beyond the familiar triad of formalism, intuitionism, and logicism, show how deeply we can see with the help of a guide determined to present fundamental mathematical ideas in ways that match our human capacities, will be of interest to graduate students and researchers in logic and the foundations of mathematics.