Representation and Productive Ambiguity in Mathematics and the Sciences

by Emily R. Grosholz

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 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.

Product Details

ISBN-13:	9780191538513
Publisher:	Oxford University Press
Publication date:	08/30/2007
Sold by:	Barnes & Noble
Format:	eBook
File size:	3 MB

Table of Contents

Preface     xi
Acknowledgments     xv
Introductory Chapters     1
Productive Ambiguity: Galileo contra Carnap     3
Galileo's Demonstration of Projectile Motion     5
Carnap on Language and Thought     16
From the Syntactic to the Semantic to the Pragmatic Approach     19
A Pragmatic Account of Berzelian Formulas     24
Analysis and Experience     33
Analysis     33
Mathematical Experience     47
Chemistry and Geometry     61
Bioorganic Chemistry and Biology     63
What Lies Between Representing and Intervening     64
The Reduction of a Biological Item to a Chemical Item     68
Formulating the Problem     71
Constructing the Antibody Mimic     74
Testing the Antibody Mimic     84
Conclusions     89
Genetics and Molecular Biology     91
Objections to Hempel's Model of Theory Reduction     92
The Transposition of Genes: McClintock and Fedoroff     97
McClintock's Studies of Maize     103
J. D. Watson's Textbook     111
Fedoroff's Translation of McClintock     113
The Future of Molecular Biology     123
Chemistry, Quantum Mechanics, and Group Theory     126
Symbols, Icons, and Iconicity     126
Representation Theory     128
Molecules, Symmetry, and Groups     131
Symmetry Groups, Representations, and Character Tables     135
The Benzene Ring and Carbocyclic Systems     139
Measuring Delocalization Energy in the Benzene Molecule     147
Geometry and Seventeeth Century Mechanics     157
Descartes's Geometry     159
Locke's Criticism of Syllogistic     159
Descartes' Geometry as the Exemplar of Cartesian Method     165
Diagrams as Procedures     169
Generalization to the Construction of a Locus     171
Generalization to Higher Algebraic Curves     177
Newton's Principia     184
Philip Kitcher on History     187
Jean Cavailles on History     189
Book I, Propositions I and VI in Newton's Principia     192
Book I, Proposition XI in Newton's Principia     198
Leibniz on Transcendental Curves     204
The Principle of Continuity     205
Studies for the Infinitesimal Calculus     207
The Principle of Perfection     213
The Isochrone and the Tractrix     215
The Catenary or La Chainette     221
Geometry and Twentieth Century Topology     225
Geometry, Algebra, and Topology     227
Vuillemin on the Relation of Mathematics and Philosophy     227
Euclid's Elements and Descartes' Geometry Revisited     230
Kant's Transcendental Aesthetic: Extrinsic and Intrinsic Intuition     232
The First Pages of Singer and Thorpe     235
De Rham's Theorem     243
Nancy Cartwright on the Abstract and Concrete     254
Logic and Topology     257
Penelope Maddy on Set Theory     259
A Brief Reconsideration of Arithmetic     262
The Application of Logic to General Topology     268
Logical Hierarchies and the Borel Hierarchy     273
Model Theory and Topological Logics     279
Coda     283
List of Illustrations     285
Glossary     291
Bibliography     293
Books     293
Articles     299
Index     307

Editorial Reviews

"She has given us a very readable and well-argued series of case studies in the history and philosophy of the sciences. Anyone with an interest in the history of mathematics, the history of science, or the general philosophy of science is bound to find a wealth of interesting material here."—Doug Jesseph, Notre Dame Philosophical Reviews

From the Publisher