The Problem

Penelope Maddy

in Defending the Axioms

Published in print January 2011 | ISBN: 9780199596188
Published online May 2011 | e-ISBN: 9780191725395 | DOI:
The Problem

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Proper understanding of contemporary set theory requires a clear sense of how it came to be what it is and the mathematical roles it is designed to play. This chapter traces the rise of pure mathematics as it gradually separated from natural science, and highlights the methodological uncertainties that present themselves when mathematics is no longer guided by empirical needs. Set theory developed at the turn of the 19th to the 20th centuries as a foundational context for determining which mathematical structures and proof techniques are legitimate. The problem then becomes: what are the proper methods of set theory, and why are these the proper methods?

Keywords: applied mathematics; foundation for mathematics; set-theoretic methods; pure mathematics; set theory

Chapter.  14050 words. 

Subjects: Philosophy of Mathematics and Logic ; Metaphysics

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