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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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PRINTED FROM PRINCETON SCHOLARSHIP ONLINE (www.princeton.universitypressscholarship.com). (c) Copyright Princeton University Press, 2021. All Rights Reserved. An individual user may print out a PDF of a single chapter of a monograph in PRSO for personal use.date: 19 September 2021

Advanced Math

Advanced Math

The Hypothetical Conception

(p.153) 6 Advanced Math
Mathematical Knowledge and the Interplay of Practices

José Ferreirós

Princeton University Press

This chapter proposes the idea that advanced mathematics is based on hypotheses—that far from being a priori, it is based on hypothetical assumptions. The concept of quasi-empiricism is often linked with the view that inductive methods are at play when the hypotheses are established. The presence of hypotheses at the very heart of mathematics establishes an important similitude with physical theory and undermines the simple distinction between “formal” and “empirical” sciences. The chapter first elaborates on a hypothetical conception of mathematics before discussing the ideas (and ideals) of certainty and objectivity in mathematics. It then considers the modern problems of the continuum that exist in ancient Greek geometry, along with the so-called methodological platonism of modern mathematics and its focus on mathematical objects. Finally, it describes the Axiom of Completeness and the Riemann Hypothesis.

Keywords:   hypotheses, advanced mathematics, certainty, objectivity, continuum, Greek geometry, methodological platonism, mathematical objects, Axiom of Completeness, Riemann Hypothesis

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