*Leon Horsten and Philip Welch (eds)*

- Published in print:
- 2016
- Published Online:
- November 2016
- ISBN:
- 9780198759591
- eISBN:
- 9780191820373
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198759591.001.0001
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy

The logician Kurt Gödel in 1951 established a disjunctive thesis about the scope and limits of mathematical knowledge: either the mathematical mind is equivalent to a Turing machine (i.e., a ...
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The logician Kurt Gödel in 1951 established a disjunctive thesis about the scope and limits of mathematical knowledge: either the mathematical mind is equivalent to a Turing machine (i.e., a computer) or there are absolutely undecidable mathematical problems. In the second half of the twentieth century, attempts have been made to arrive at a stronger conclusion. In particular, arguments have been produced by the philosopher J.R. Lucas and by the physicist and mathematician Roger Penrose that intend to show that the mathematicalmind ismore powerful than any computer. These arguments, and counterarguments to them, have not convinced the logical and philosophical community. The reason for this is an insufficiency of rigour in the debate. The contributions in this volume move the debate forward by formulating rigorous frameworks and formally spelling out and evaluating arguments that bear on Gödel’s disjunction in these frameworks. The contributions in this volume have been written by world leading experts in the field.Less

The logician Kurt Gödel in 1951 established a disjunctive thesis about the scope and limits of mathematical knowledge: either the mathematical mind is equivalent to a Turing machine (i.e., a computer) or there are absolutely undecidable mathematical problems. In the second half of the twentieth century, attempts have been made to arrive at a stronger conclusion. In particular, arguments have been produced by the philosopher J.R. Lucas and by the physicist and mathematician Roger Penrose that intend to show that the mathematicalmind ismore powerful than any computer. These arguments, and counterarguments to them, have not convinced the logical and philosophical community. The reason for this is an insufficiency of rigour in the debate. The contributions in this volume move the debate forward by formulating rigorous frameworks and formally spelling out and evaluating arguments that bear on Gödel’s disjunction in these frameworks. The contributions in this volume have been written by world leading experts in the field.

*Selman Akbulut*

- Published in print:
- 2016
- Published Online:
- November 2016
- ISBN:
- 9780198784869
- eISBN:
- 9780191827136
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198784869.001.0001
- Subject:
- Mathematics, Geometry / Topology

This book present the topology of smooth 4-manifolds in an intuitive self-contained way. The handlebody theory, and the seiberg-witten theory of 4-manifolds are presented. Also stein and symplectic ...
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This book present the topology of smooth 4-manifolds in an intuitive self-contained way. The handlebody theory, and the seiberg-witten theory of 4-manifolds are presented. Also stein and symplectic structures on 4-manifolds are discussed, and many recent applications are given.Less

This book present the topology of smooth 4-manifolds in an intuitive self-contained way. The handlebody theory, and the seiberg-witten theory of 4-manifolds are presented. Also stein and symplectic structures on 4-manifolds are discussed, and many recent applications are given.

*Andreas J. Stylianides*

- Published in print:
- 2016
- Published Online:
- September 2016
- ISBN:
- 9780198723066
- eISBN:
- 9780191789588
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198723066.001.0001
- Subject:
- Mathematics, Educational Mathematics

Proving in the Elementary Mathematics Classroom addresses a fundamental problem in children’s learning that has received relatively little research attention: Although proving and related concepts ...
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Proving in the Elementary Mathematics Classroom addresses a fundamental problem in children’s learning that has received relatively little research attention: Although proving and related concepts (e.g., proof, argumentation, conjecturing) are core to mathematics as a sense-making activity, they currently have a marginal place in elementary classrooms internationally. This book takes a step toward addressing this problem by examining how the place of proving in elementary students’ mathematical work can be elevated through the purposeful design and implementation of mathematics tasks, specifically proving tasks. In particular, the book draws on relevant research and theory and classroom episodes with 8–9-year-olds from England and the United States to examine different kinds of proving tasks and the proving activity they can help generate in the elementary classroom. It examines further the role of elementary teachers in mediating the relationship between proving tasks and proving activity, including major mathematical and pedagogical issues that can arise for them as they implement each kind of proving task in the classroom. In addition to its research contribution in the intersection of the scholarly areas of teaching/learning proving and task design/implementation, the book has important implications for teaching, curricular resources, and teacher education. For example, the book identifies different kinds of proving tasks whose balanced representation in the mathematics classroom and in curricular resources can support a rounded set of learning experiences for elementary students related to proving. It identifies further important mathematical ideas and pedagogical practices related to proving that can be studied in teacher education.Less

*Proving in the Elementary Mathematics Classroom* addresses a fundamental problem in children’s learning that has received relatively little research attention: Although proving and related concepts (e.g., proof, argumentation, conjecturing) are core to mathematics as a sense-making activity, they currently have a marginal place in elementary classrooms internationally. This book takes a step toward addressing this problem by examining how the place of proving in elementary students’ mathematical work can be elevated through the purposeful design and implementation of mathematics tasks, specifically proving tasks. In particular, the book draws on relevant research and theory and classroom episodes with 8–9-year-olds from England and the United States to examine different kinds of proving tasks and the proving activity they can help generate in the elementary classroom. It examines further the role of elementary teachers in mediating the relationship between proving tasks and proving activity, including major mathematical and pedagogical issues that can arise for them as they implement each kind of proving task in the classroom. In addition to its research contribution in the intersection of the scholarly areas of teaching/learning proving and task design/implementation, the book has important implications for teaching, curricular resources, and teacher education. For example, the book identifies different kinds of proving tasks whose balanced representation in the mathematics classroom and in curricular resources can support a rounded set of learning experiences for elementary students related to proving. It identifies further important mathematical ideas and pedagogical practices related to proving that can be studied in teacher education.