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Philosophy of MathematicsStructure and Ontology$
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Stewart Shapiro

Print publication date: 2000

Print ISBN-13: 9780195139303

Published to Oxford Scholarship Online: November 2003

DOI: 10.1093/0195139305.001.0001

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How We Got Here

How We Got Here

(p.143) 5 How We Got Here
Philosophy of Mathematics

Stewart Shapiro (Contributor Webpage)

Oxford University Press

This chapter sketches the historical development, within mathematics, of the idea that mathematics is the science of structure. We begin with the complex transition from geometry as the study of physical or perceived space to geometry as the study of freestanding structures. Of particular note is the debate between Poincaré and Russell, and the debate between Frege and Hilbert. Hilbert's work on geometry is the culmination of the programme to banish anything like Kantian intuition, in favour of implicit definitions. Another crucial event was Dedekind's work in arithmetic and real analysis. The Bourbaki school is also briefly investigated.

Keywords:   Bourbaki, Dedekind, Frege, geometry, Hilbert, intuition, Kant, Poincaré, Russell, structure

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