Representation and Productive Ambiguity in Mathematics and the SciencesClarendon Press, 2007 M08 30 - 330 páginas 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. |
Contenido
Chemistry and Geometry | 61 |
Geometry and Seventeenth Century Mechanics | 157 |
Geometry and Twentieth Century Topology | 225 |
List of Illustrations | 285 |
Glossary | 291 |
293 | |
307 | |
Otras ediciones - Ver todas
Representation and Productive Ambiguity in Mathematics and the Sciences Emily R. Grosholz,Emily Grosholz Vista previa limitada - 2007 |
Representation and Productive Ambiguity in Mathematics and the Sciences Emily R. Grosholz Sin vista previa disponible - 2007 |
Términos y frases comunes
abstract algebraic ambiguity analysis antibody mimic arithmetic articulation atomic atomic orbitals benzene Berzelian formulas Borel hierarchy Calixarene canonical Carnap Cartesian chapter chemistry chemists complex conditions of intelligibility construction curves Descartes diagram differential domain Elementary Topology equation Euclidean example exhibit Fedoroff Figure finite formal function Galileo gene genetics group theory hierarchy iconic representations infinitesimal intuition irreducible isomorphism knowledge Lecture Notes Leibniz line segments locus macroscopic Maize manifold mathematical mathematicians McClintock means methods modes of representation molecular biology molecular orbitals molecule natural language natural numbers Newton notation Notes on Elementary Nucleotide Sequence objects ofthe open sets paper tools philosophical philosophy of mathematics predicate logic problems Proposition rational relation real numbers recursive reduction represent Reprinted with kind role set theory simplicial complex Singer and Thorpe solve structure symbolic symmetry Theorem topological space Topology and Geometry triangle valence bond theory Vuillemin