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Mathematical knowledge and the interplay of practices / Jos�e Ferreir�os.

By: Ferreir�os Dom�inguez, Jos�e [author.].
Material type: materialTypeLabelBookPublisher: Princeton : Princeton University Press, 2015Description: 1 online resource.Content type: text Media type: computer Carrier type: online resourceISBN: 9781400874002; 1400874009.Subject(s): Mathematics -- Philosophy | Math�ematiques -- Philosophie | MATHEMATICS -- Essays | MATHEMATICS -- Pre-Calculus | MATHEMATICS -- Reference | MATHEMATICS -- General | Mathematics -- PhilosophyGenre/Form: Electronic books.Additional physical formats: Print version:: Ferreir�os, Jos�e. Mathematical Knowledge and the Interplay of PracticesDDC classification: 510.1 Online resources: Click here to access online
Contents:
Cover -- Contents -- List of Illustrations -- Foreword -- 1 On Knowledge and Practices: A Manifesto -- 2 The Web of Practices -- 2.1. Historical Work on Practices -- 2.2. Philosophers Working on Practices -- 2.3. What Is Mathematical Practice, Then? -- 2.4. The Multiplicity of Practices -- 2.5. The Interplay of Practices and Its Basis -- 3 Agents and Frameworks -- 3.1. Frameworks and Related Matters -- 3.2. Interlude on Examplars -- 3.3. On Agents -- 3.4. Counting Practices and Cognitive Abilities -- 3.5. Further Remarks on Mathematics and Cognition -- 3.6. Agents and "Metamathematical" Views -- 3.7. On Systematic Links -- 4 Complementarity in Mathematics -- 4.1. Formula and Meaning -- 4.2. Formal Systems and Intended Models -- 4.3. Meaning in Mathematics: A Tentative Approach -- 4.4. The Case of Complex Numbers -- 5 Ancient Greek Mathematics: A Role for Diagrams -- 5.1. From the Technical to the Mathematical -- 5.2. The Elements: Getting Started -- 5.3. On the Euclidean Postulates: Ruling Diagrams (and Their Reading) -- 5.4. Diagram-Based Mathematics and Proofs -- 5.5. Agents, Idealization, and Abstractness -- 5.6. A Look at the Future-Our Past -- 6 Advanced Math: The Hypothetical Conception -- 6.1. The Hypothetical Conception: An Introduction -- 6.2. On Certainty and Objectivity -- 6.3. Elementary vs. Advanced: Geometry and the Continuum -- 6.4. Talking about Objects -- 6.5. Working with Hypotheses: AC and the Riemann Conjecture -- 7 Arithmetic Certainty -- 7.1. Basic Arithmetic -- 7.2. Counting Practices, Again -- 7.3. The Certainty of Basic Arithmetic -- 7.4. Further Clarifications -- 7.5. Model Theory of Arithmetic -- 7.6. Logical Issues: Classical or Intuitionistic Math? -- 8 Mathematics Developed: The Case of the Reals -- 8.1. Inventing the Reals -- 8.2. "Tenths" to the Infinite: Lambert and Newton -- 8.3. The Number Continuum.
8.4. The Reinvention of the Reals -- 8.5. Simple Infinity and Arbitrary Infinity -- 8.6. Developing Mathematics -- 8.7. Mathematical Hypotheses and Scientific Practices -- 9 Objectivity in Mathematical Knowledge -- 9.1. Objectivity and Mathematical Hypotheses: A Simple Case -- 9.2. Cantor's "Purely Arithmetical" Proofs -- 9.3. Objectivity and Hypotheses, II: The Case of p(N) -- 9.4. Arbitrary Sets and Choice -- 9.5. What about Cantor's Ordinal Numbers? -- 9.6. Objectivity and the Continuum Problem -- 10 The Problem of Conceptual Understanding -- 10.1. The Universe of Sets -- 10.2. A "Web-of-Practices" Look at the Cumulative Picture -- 10.3. Conceptual Understanding -- 10.4. Justifying Set Theory: Arguments Based on the Real-Number Continuum -- 10.5. By Way of Conclusion -- References -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- U -- V -- W -- Z.
Summary: Annotation This book presents a new approach to the epistemology of mathematics by viewing mathematics as a human activity whose knowledge is intimately linked with practice. Charting an exciting new direction in the philosophy of mathematics, Jose Ferreiros uses the crucial idea of a continuum to provide an account of the development of mathematical knowledge that reflects the actual experience of doing math and makes sense of the perceived objectivity of mathematical results.Describing a historically oriented, agent-based philosophy of mathematics, Ferreiros shows how the mathematical tradition evolved from Euclidean geometry to the real numbers and set-theoretic structures. He argues for the need to take into account a whole web of mathematical and other practices that are learned and linked by agents, and whose interplay acts as a constraint. Ferreiros demonstrates how advanced mathematics, far from being a priori, is based on hypotheses, in contrast to elementary math, which has strong cognitive and practical roots and therefore enjoys certainty.Offering a wealth of philosophical and historical insights, "Mathematical Knowledge and the Interplay of Practices" challenges us to rethink some of our most basic assumptions about mathematics, its objectivity, and its relationship to culture and science."
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Includes bibliographical references and index.

Online resource; title from PDF title page (Ebsco, viewed on November 19, 2015).

Cover -- Contents -- List of Illustrations -- Foreword -- 1 On Knowledge and Practices: A Manifesto -- 2 The Web of Practices -- 2.1. Historical Work on Practices -- 2.2. Philosophers Working on Practices -- 2.3. What Is Mathematical Practice, Then? -- 2.4. The Multiplicity of Practices -- 2.5. The Interplay of Practices and Its Basis -- 3 Agents and Frameworks -- 3.1. Frameworks and Related Matters -- 3.2. Interlude on Examplars -- 3.3. On Agents -- 3.4. Counting Practices and Cognitive Abilities -- 3.5. Further Remarks on Mathematics and Cognition -- 3.6. Agents and "Metamathematical" Views -- 3.7. On Systematic Links -- 4 Complementarity in Mathematics -- 4.1. Formula and Meaning -- 4.2. Formal Systems and Intended Models -- 4.3. Meaning in Mathematics: A Tentative Approach -- 4.4. The Case of Complex Numbers -- 5 Ancient Greek Mathematics: A Role for Diagrams -- 5.1. From the Technical to the Mathematical -- 5.2. The Elements: Getting Started -- 5.3. On the Euclidean Postulates: Ruling Diagrams (and Their Reading) -- 5.4. Diagram-Based Mathematics and Proofs -- 5.5. Agents, Idealization, and Abstractness -- 5.6. A Look at the Future-Our Past -- 6 Advanced Math: The Hypothetical Conception -- 6.1. The Hypothetical Conception: An Introduction -- 6.2. On Certainty and Objectivity -- 6.3. Elementary vs. Advanced: Geometry and the Continuum -- 6.4. Talking about Objects -- 6.5. Working with Hypotheses: AC and the Riemann Conjecture -- 7 Arithmetic Certainty -- 7.1. Basic Arithmetic -- 7.2. Counting Practices, Again -- 7.3. The Certainty of Basic Arithmetic -- 7.4. Further Clarifications -- 7.5. Model Theory of Arithmetic -- 7.6. Logical Issues: Classical or Intuitionistic Math? -- 8 Mathematics Developed: The Case of the Reals -- 8.1. Inventing the Reals -- 8.2. "Tenths" to the Infinite: Lambert and Newton -- 8.3. The Number Continuum.

8.4. The Reinvention of the Reals -- 8.5. Simple Infinity and Arbitrary Infinity -- 8.6. Developing Mathematics -- 8.7. Mathematical Hypotheses and Scientific Practices -- 9 Objectivity in Mathematical Knowledge -- 9.1. Objectivity and Mathematical Hypotheses: A Simple Case -- 9.2. Cantor's "Purely Arithmetical" Proofs -- 9.3. Objectivity and Hypotheses, II: The Case of p(N) -- 9.4. Arbitrary Sets and Choice -- 9.5. What about Cantor's Ordinal Numbers? -- 9.6. Objectivity and the Continuum Problem -- 10 The Problem of Conceptual Understanding -- 10.1. The Universe of Sets -- 10.2. A "Web-of-Practices" Look at the Cumulative Picture -- 10.3. Conceptual Understanding -- 10.4. Justifying Set Theory: Arguments Based on the Real-Number Continuum -- 10.5. By Way of Conclusion -- References -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- U -- V -- W -- Z.

Annotation This book presents a new approach to the epistemology of mathematics by viewing mathematics as a human activity whose knowledge is intimately linked with practice. Charting an exciting new direction in the philosophy of mathematics, Jose Ferreiros uses the crucial idea of a continuum to provide an account of the development of mathematical knowledge that reflects the actual experience of doing math and makes sense of the perceived objectivity of mathematical results.Describing a historically oriented, agent-based philosophy of mathematics, Ferreiros shows how the mathematical tradition evolved from Euclidean geometry to the real numbers and set-theoretic structures. He argues for the need to take into account a whole web of mathematical and other practices that are learned and linked by agents, and whose interplay acts as a constraint. Ferreiros demonstrates how advanced mathematics, far from being a priori, is based on hypotheses, in contrast to elementary math, which has strong cognitive and practical roots and therefore enjoys certainty.Offering a wealth of philosophical and historical insights, "Mathematical Knowledge and the Interplay of Practices" challenges us to rethink some of our most basic assumptions about mathematics, its objectivity, and its relationship to culture and science."

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