The Foundations of Mathematics

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Title:	The Foundations of Mathematics
Description:	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.
Authors:	Ian Stewart, David Tall
Resource Type:	eBook.
Subjects:	Logic, Symbolic and mathematical
Categories:	MATHEMATICS / Number Systems
Database:	eBook Collection (EBSCOhost)

FullText	Links: – Type: ebook-pdf Text: Availability: 0
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Items	– Name: Title Label: Title Group: Ti Data: The Foundations of Mathematics – Name: Abstract Label: Description Group: Ab Data: 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. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Ian+Stewart%22">Ian Stewart</searchLink><br /><searchLink fieldCode="AR" term="%22David+Tall%22">David Tall</searchLink> – Name: TypePub Label: Resource Type Group: TypPub Data: eBook. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Logic%2C+Symbolic+and+mathematical%22">Logic, Symbolic and mathematical</searchLink> – Name: SubjectBISAC Label: Categories Group: Su Data: <searchLink fieldCode="ZK" term="%22MATHEMATICS+%2F+Number+Systems%22">MATHEMATICS / Number Systems</searchLink>
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RecordInfo	BibRecord: BibEntity: Classifications: – Code: 511.3 Scheme: ddc Type: prePub Languages: – Code: eng Text: English Subjects: – SubjectFull: Logic, Symbolic and mathematical Type: general Titles: – TitleFull: The Foundations of Mathematics Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Ian Stewart – PersonEntity: Name: NameFull: David Tall – PersonEntity: Name: NameFull: Ian Stewart – PersonEntity: Name: NameFull: David Tall IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 01 Type: published Y: 2015 – D: 13 M: 02 Type: profile Y: 2015 Identifiers: – Type: isbn-print Value: 9780198706434 – Type: isbn-print Value: 9780191016486 – Type: isbn-print Value: 9780198706441 – Type: isbn-electronic Value: 9780191016479 Titles: – TitleFull: The Foundations of Mathematics Type: main
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