The Foundations of Mathematics

Filename: the-foundations-of-mathematics.pdf
ISBN: 9780191016486
Release Date: 2015-03-12
Number of pages: 432
Author: Ian Stewart
Publisher: OUP Oxford

Download and read online The Foundations of Mathematics in PDF and EPUB 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.


Wittgenstein s Lectures on the Foundations of Mathematics Cambridge 1939

Filename: wittgenstein-s-lectures-on-the-foundations-of-mathematics-cambridge-1939.pdf
ISBN: 0226904261
Release Date: 1989-10-15
Number of pages: 300
Author: Ludwig Wittgenstein
Publisher: University of Chicago Press

Download and read online Wittgenstein s Lectures on the Foundations of Mathematics Cambridge 1939 in PDF and EPUB 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.


The Foundations of Mathematics

Filename: the-foundations-of-mathematics.pdf
ISBN: 1904987141
Release Date: 2009
Number of pages: 251
Author: Kenneth Kunen
Publisher:

Download and read online The Foundations of Mathematics in PDF and EPUB 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.


Introduction to the Foundations of Mathematics

Filename: introduction-to-the-foundations-of-mathematics.pdf
ISBN: 9780486276205
Release Date: 2013-09-26
Number of pages: 352
Author: Raymond L. Wilder
Publisher: Courier Corporation

Download and read online Introduction to the Foundations of Mathematics in PDF and EPUB This classic undergraduate text acquaints students with the fundamental concepts and methods of mathematics. In addition to introducing many historical figures from the 18th through the mid-20th centuries, it examines the axiomatic method, set theory, infinite sets, groups, intuitionism, formal systems, mathematical logic, and other topics. 1965 second edition.


Foundations of Mathematics and Other Logical Essays

Filename: foundations-of-mathematics-and-other-logical-essays.pdf
ISBN: 9781134528035
Release Date: 2013-10-15
Number of pages: 312
Author: Frank Plumpton Ramsey
Publisher: Routledge

Download and read online Foundations of Mathematics and Other Logical Essays in PDF and EPUB First published in 2000. Routledge is an imprint of Taylor & Francis, an informa company.


Wittgenstein Finitism and the Foundations of Mathematics

Filename: wittgenstein-finitism-and-the-foundations-of-mathematics.pdf
ISBN: 019823516X
Release Date: 1998
Number of pages: 260
Author: Mathieu Marion
Publisher: Oxford University Press

Download and read online Wittgenstein Finitism and the Foundations of Mathematics in PDF and EPUB 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.


Essays on the Foundations of Mathematics and Logic

Filename: essays-on-the-foundations-of-mathematics-and-logic.pdf
ISBN: 9788876990144
Release Date: 2005-01-01
Number of pages: 351
Author: Giandomenico Sica
Publisher: Polimetrica s.a.s.

Download and read online Essays on the Foundations of Mathematics and Logic in PDF and EPUB


Kurt G del and the Foundations of Mathematics

Filename: kurt-g-del-and-the-foundations-of-mathematics.pdf
ISBN: 9781139498432
Release Date: 2011-06-06
Number of pages:
Author: Matthias Baaz
Publisher: Cambridge University Press

Download and read online Kurt G del and the Foundations of Mathematics in PDF and EPUB 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.


Introduction to the Foundations of Mathematics

Filename: introduction-to-the-foundations-of-mathematics.pdf
ISBN: 9780486276205
Release Date: 2013-09-26
Number of pages: 352
Author: Raymond L. Wilder
Publisher: Courier Corporation

Download and read online Introduction to the Foundations of Mathematics in PDF and EPUB This classic undergraduate text acquaints students with the fundamental concepts and methods of mathematics. In addition to introducing many historical figures from the 18th through the mid-20th centuries, it examines the axiomatic method, set theory, infinite sets, groups, intuitionism, formal systems, mathematical logic, and other topics. 1965 second edition.


The Foundations of Mathematics

Filename: the-foundations-of-mathematics.pdf
ISBN: 9780470085011
Release Date: 2008-04-07
Number of pages: 392
Author: Thomas Q. Sibley
Publisher: John Wiley & Sons

Download and read online The Foundations of Mathematics in PDF and EPUB 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.


The Foundations of Mathematics in the Theory of Sets

Filename: the-foundations-of-mathematics-in-the-theory-of-sets.pdf
ISBN: 0521770343
Release Date: 2000
Number of pages: 424
Author: John P. Mayberry
Publisher: Cambridge University Press

Download and read online The Foundations of Mathematics in the Theory of Sets in PDF and EPUB This 2001 book will appeal to mathematicians and philosophers interested in the foundations of mathematics.


Practical Foundations of Mathematics

Filename: practical-foundations-of-mathematics.pdf
ISBN: 0521631076
Release Date: 1999-05-13
Number of pages: 572
Author: Paul Taylor
Publisher: Cambridge University Press

Download and read online Practical Foundations of Mathematics in PDF and EPUB 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.



Remarks on the Foundations of Mathematics

Filename: remarks-on-the-foundations-of-mathematics.pdf
ISBN: 0631125051
Release Date: 1981-03
Number of pages: 448
Author: Ludwig Wittgenstein
Publisher: Wiley-Blackwell

Download and read online Remarks on the Foundations of Mathematics in PDF and EPUB This substantially revised edition of Wittgenstein′s Remarks on the Foundations of Mathematics contains one section, an essay of fifty pages, not previously published, as well as considerable additions to others sections. In Parts I, II and III, Wittgenstein discusses amongst other things the idea that all strict reasoning, and so all mathematics, are built on the ′fundamental calculus′ which is logic. These parts give the most thorough discussion of Russell′s logic. He writes on mathematical proof and the question of where the proofs of mathematics get their force and cogency, if they are not reducible to proofs in logic. Thsi leads him to discuss′contradiction in mathematics′ and ′consistency proofs′. He works against the view that there is a sharp division between ′grammatical propositions′ and ′empirical prepositions′. He asks us at one point to imagine a people who made no distinction between the applied mathematics and pure mathematics, although they counted and calculated. Could we say they had proofs? Here is a feature of his method which becomes more imporatnt; what Wittgenstein calls, at least half seriously, ′the anthropological method in philosophy′. This emerges in Parts V, VI and VIII. In Part VI, published here for the first time, Wittgenstein brings togeher the view that in mathematics proofs ae ′concept forming′ and the view that language and logic and mathematics ′presuppose′ common ways of acting and of living among the people who give tham and are convinced by them. Part VIII now has a fuller discussion of difficulties in the notion of ′following a rule′ in calculation and the notion of logical necessity.


The Foundations of Mathematics

Filename: the-foundations-of-mathematics.pdf
ISBN: 9780791097212
Release Date: 2006-01-01
Number of pages:
Author: Michael John Bradley
Publisher: Infobase Publishing

Download and read online The Foundations of Mathematics in PDF and EPUB During the 16th and 17th centuries, mathematicians developed a wealth of new ideas but had not carefully employed accurate definitions, proofs, or procedures to document and implement them. However, in the early 19th century, mathematicians began to recognize the need to precisely define their terms, to logically prove even obvious principles, and to use rigorous methods of manipulation. The Foundations of Mathematics presents the lives and accomplishments of 10 mathematicians who lived between CE 1800 and 1900 and contributed to one or more of the four major initiatives that characterized the rapid growth of mathematics during the 19th century: the introduction of rigor, the investigation of the structure of mathematical systems, the development of new branches of mathematics, and the spread of mathematical activity throughout Europe. This readable new volume communicates the importance and impact of the work of the pioneers who redefined this area of study.