Representation and Productive Ambiguity in Mathematics and the Sciences - Hardcover

Grosholz, Emily R.

Hardcover

ISBN 10: 0199299730 ISBN 13: 9780199299737

Publisher: OUP Oxford, 2007

Synopsis

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.

About this title

Review

The wide range of topics and detail in which they are discussed offer some significant insights into the philosophy of science and mathematics. Moreover, Grosholz makes a generally persuasive case that scientific progress is often the result of a synthesis from several traditions that combines very different ways of representing the same subject matter... 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)

Grosholz has done a good job of laying out some of the more fascinating contents of the scientists' and mathematicians' tool boxes (Steven French, British Journal of the Philosophy of Science)

Anyone interested in mathematical practice will find plenty of material in this book ... philosophers who wish to reject Grosholz's views had better come up with collections of studies at least as detailed and compelling as these. That is no small challenge. (Brendan P. Larvor, Philosophia Mathematica)

Edition details

• Publisher: OUP Oxford
• Publication date: 2007
• ISBN 10: 0199299730
• ISBN 13: 9780199299737
• Binding: Hardcover
• Language: English
• Number of pages: 332