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Mathematical Knowledge and the Interplay of Practices$
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José Ferreirós

Print publication date: 2015

Print ISBN-13: 9780691167510

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691167510.001.0001

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The Problem of Conceptual Understanding

The Problem of Conceptual Understanding

Chapter:
(p.281) 10 The Problem of Conceptual Understanding
Source:
Mathematical Knowledge and the Interplay of Practices
Author(s):

José Ferreirós

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691167510.003.0010

This chapter considers one of the most intriguing questions that philosophy of mathematics in practice must, sooner or later, confront: how understanding of mathematics is obtained. In particular, it examines how issues of meaning and understanding in relation to practice and use relate to the question of the acceptability of “classical” or postulational mathematics, a question usually formulated in terms of consistency. The chapter begins with a discussion of the iterative conception of the universe of sets and its presuppositions, analyzing it from the standpoint of the web of practices. It then addresses the issue of conceptual understanding in mathematics, as exemplifid by the theory Zermelo–Fraenkel axiom system (ZFC). Finally, it looks at arguments based on the idea of the real-number continuum as a source of justification for the axioms of set theory.

Keywords:   practice, postulational mathematics, sets, conceptual understanding, continuum, axioms, set theory, Zermelo–Fraenkel axiom system

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