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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