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The Foundations of Mathematics

Author: Ian Stewart

Publisher: Oxford University Press, USA

ISBN: 9780198706434

Category: Mathematics

Page: 432

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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.
The Foundations of Mathematics
Language: en
Pages: 432
Authors: Ian Stewart, David Tall
Categories: Mathematics
Type: BOOK - Published: 2015-03-05 - Publisher: Oxford University Press, USA

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
The Foundations of Mathematics
Language: en
Pages: 392
Authors: Thomas Q. Sibley
Categories: Mathematics
Type: BOOK - Published: 2008-04-07 - Publisher: John Wiley & Sons

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
The Foundations of Mathematics and Other Logical Essays
Language: en
Pages: 292
Authors: Frank Plumpton Ramsey
Categories: Philosophy
Type: BOOK - Published: 2000 - Publisher: Psychology Press

First published in 2000. Routledge is an imprint of Taylor & Francis, an informa company.
Wittgenstein, Finitism, and the Foundations of Mathematics
Language: en
Pages: 260
Authors: Assistant Professor of Philosophy Mathieu Marion, Mathieu Marion
Categories: Mathematics
Type: BOOK - Published: 1998 - Publisher: Oxford University Press

This is a careful, historically informed study of Wittgenstein's philosophy of mathematics, tracing the work development of his thinking from the 1920s through to the 1950s, in the context of the mathematical and philosophical work of the times.
Kurt Gödel and the Foundations of Mathematics
Language: en
Pages:
Authors: Matthias Baaz, Christos H. Papadimitriou, Hilary W. Putnam, Dana S. Scott, Charles L. Harper, Jr
Categories: Mathematics
Type: BOOK - Published: 2011-06-06 - Publisher: Cambridge University Press

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