Toward a clarity of the extreme value theorem

TL;DR

This paper uses Peirce's framework to analyze and compare intuitionist and infinitesimal approaches to the extreme value theorem, highlighting the complementarity of different mathematical perspectives.

Contribution

It introduces a philosophical analysis of mathematical clarity, comparing two approaches to a fundamental theorem, emphasizing their complementary nature rather than rivalry.

Findings

Different approaches capture different aspects of the theorem

Mathematical phenomena may have multiple valid formalizations

Complementarity enhances understanding of analysis concepts

Abstract

We apply a framework developed by C. S. Peirce to analyze the concept of clarity, so as to examine a pair of rival mathematical approaches to a typical result in analysis. Namely, we compare an intuitionist and an infinitesimal approaches to the extreme value theorem. We argue that a given pre-mathematical phenomenon may have several aspects that are not necessarily captured by a single formalisation, pointing to a complementarity rather than a rivalry of the approaches.