Representation and Productive Ambiguity in Mathematics and the Sciences / Edition 1

Representation and Productive Ambiguity in Mathematics and the Sciences / Edition 1

by Emily R. Grosholz
     
 

ISBN-10: 0199299730

ISBN-13: 9780199299737

Pub. Date: 10/18/2007

Publisher: Oxford University Press

Emily Grosholz offers an original investigation of demonstration in mathematics and science, examining how it works and why it is persuasive. Focusing on geometrical demonstration, she shows the roles that representation and ambiguity play in mathematical discovery. She presents a wide range of case studies in mechanics, topology, algebra, logic, and chemistry,

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Overview

Emily Grosholz offers an original investigation of demonstration in mathematics and science, examining how it works and why it is persuasive. Focusing on geometrical demonstration, she shows the roles that representation and ambiguity play in mathematical discovery. She presents a wide range of case studies in mechanics, topology, algebra, logic, and chemistry, from ancient Greece to the present day, but focusing particularly on the seventeenth and twentieth centuries. She argues that reductive methods are effective not because they diminish but because they multiply and juxtapose modes of representation. Such problem-solving is, she argues, best understood in terms of Leibnizian "analysis"—the search for conditions of intelligibility. Discovery and justification are then two aspects of one rational way of proceeding, which produces the mathematician's formal experience.

Grosholz defends the importance of iconic, as well as symbolic and indexical, signs in mathematical representation, and argues that pragmatic, as well as syntactic and semantic, considerations are indispensable fore mathematical reasoning. By taking a close look at the way results are presented on the page in mathematical (and biological, chemical, and mechanical) texts, she shows that when two or more traditions combine in the service of problem solving, notations and diagrams are subtly altered, multiplied, and juxtaposed, and surrounded by prose in natural language which explains the novel combination. Viewed this way, the texts yield striking examples of language and notation that are irreducibly ambiguous and productive because they are ambiguous. Grosholtz's arguments, which invoke Descartes, Locke, Hume, and Kant, will be of considerable interest to philosophers and historians of mathematics and science, and also have far-reaching consequences for epistemology and philosophy of language.

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Product Details

ISBN-13:
9780199299737
Publisher:
Oxford University Press
Publication date:
10/18/2007
Pages:
332
Product dimensions:
9.30(w) x 6.30(h) x 1.00(d)

Table of Contents

Preface
Acknowledgments
Introductory Chapters
Productive Ambiguity: Galileo contra Carnap
Analysis and Experience
Section I: Chemistry and Geometry
Bioorganic Chemistry and Biology
Genetics and Molecular Biology
Representation Theory and Chemistry
Section II: Geometry and 17th century Mechanics
Descartes's Geometry
Newton's Principia
Leibniz on Transcendental Curves
Section III: Geometry and 20th century Topology
Geometry, Algebra, and Topology
Logic and Topology
Bibliography

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