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Representation and Productive Ambiguity in Mathematics and the Sciences 1st Edition by Emily R Grosholz ISBN 1435610164 9781435610163

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Author(s): Emily R. Grosholz
ISBN(s): 9781435610163, 1435610164
Edition: 1
File Details: PDF, 2.98 MB
Year: 2007
Language: english
SKU: EB-1408760 Categories: , Tag:

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ISBN 10: 1435610164 
ISBN 13: 9781435610163
Author: Emily R Grosholz

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 for 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 sublty 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.

Representation and Productive Ambiguity in Mathematics and the Sciences 1st Table of contents:

Part I: Introductory Chapters
1. Productive Ambiguity: Galileo contra Carnap
 1.1. Galileo’s Demonstration of Projectile Motion
 1.2. Carnap on Language and Thought
 1.3. From the Syntactic to the Semantic to the Pragmatic Approach
 1.4. A Pragmatic Account of Berzelian Formulas

2. Analysis and Experience
 2.1. Analysis
 2.2. Mathematical Experience

Part II: Chemistry and Geometry
3. Bioorganic Chemistry and Biology
 3.1. What Lies Between Representing and Intervening
 3.2. The Reduction of a Biological Item to a Chemical Item
 3.3. Formulating the Problem
 3.4. Constructing the Antibody Mimic
 3.5. Testing the Antibody Mimic
 3.6. Conclusions

4. Genetics and Molecular Biology
 4.1. Objections to Hempel’s Model of Theory Reduction
 4.2. The Transposition of Genes: McClintock and Fedoroff
 4.3. McClintock’s Studies of Maize
 4.4. J. D. Watson’s Textbook
 4.5. Fedoroff’s Translation of McClintock
 4.6. The Future of Molecular Biology

5. Chemistry, Quantum Mechanics, and Group Theory
 5.1. Symbols, Icons, and Iconicity
 5.2. Representation Theory
 5.3. Molecules, Symmetry, and Groups
 5.4. Symmetry Groups, Representations, and Character Tables
 5.5. The Benzene Ring and Carbocyclic Systems
 5.6. Measuring Delocalization Energy in the Benzene Molecule

Part III: Geometry and Seventeenth Century Mechanics
6. Descartes’s Geometry
 6.1. Locke’s Criticism of Syllogistic
 6.2. Descartes’ Geometry as the Exemplar of Cartesian Method
 6.3. Diagrams as Procedures
 6.4. Generalization to the Construction of a Locus
 6.5. Generalization to Higher Algebraic Curves

7. Newton’s Principia
 7.1. Philip Kitcher on History
 7.2. Jean Cavaillès on History
 7.3. Book I, Propositions I and VI in Newton’s Principia
 7.4. Book I, Proposition XI in Newton’s Principia

8. Leibniz on Transcendental Curves
 8.1. The Principle of Continuity
 8.2. Studies for the Infinitesimal Calculus
 8.3. The Principle of Perfection
 8.4. The Isochrone and the Tractrix
 8.5. The Catenary or La Chainette

Part IV: Geometry and Twentieth Century Topology
9. Geometry, Algebra, and Topology
 9.1. Vuillemin on the Relation of Mathematics and Philosophy
 9.2. Euclid’s Elements and Descartes’ Geometry Revisited
 9.3. Kant’s Transcendental Aesthetic: Extrinsic and Intrinsic Intuition
 9.4. The First Pages of Singer and Thorpe
 9.5. De Rham’s Theorem
 9.6. Nancy Cartwright on the Abstract and Concrete

10. Logic and Topology
 10.1. Penelope Maddy on Set Theory
 10.2. A Brief Reconsideration of Arithmetic
 10.3. The Application of Logic to General Topology
 10.4. Logical Hierarchies and the Borel Hierarchy
 10.5. Model Theory and Topological Logics

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Tags: Emily R Grosholz, Productive Ambiguity, Mathematics, Sciences