Weyl and intuitionistic infinitesimals

TL;DR

This paper explores the concept of infinitesimals from a Brouwerian perspective, analyzing classical and intuitionistic approaches, and suggests Reeb's approach as a promising avenue aligned with Weyl's views.

Contribution

It provides a detailed analysis of infinitesimals within Brouwer's intuitionism and proposes that Reeb's approach may best align with Weyl's philosophical stance.

Findings

Classical nonstandard analysis approaches are contrasted with Brouwerian perspectives.

Explicit Brouwerian approximations to infinitesimals have been developed by early mathematicians.

Reeb's approach is identified as potentially most suitable for Weyl's philosophical and mathematical views.

Abstract

As Weyl was interested in infinitesimal analysis and for some years embraced Brouwer's intuitionism, which he continued to see as an ideal even after he had convinced himself that it is a practical necessity for science to go beyond intuitionistic mathematics, this note presents some remarks on infinitesimals from a Brouwerian perspective. After an introduction and a look at Robinson's and Nelson's approaches to classical nonstandard analysis, three desiderata for an intuitionistic construction of infinitesimals are extracted from Brouwer's writings. These cannot be met, but in explicitly Brouwerian settings what might in different ways be called approximations to infinitesimals have been developed by early Brouwer, Vesley, and Reeb. I conclude that perhaps Reeb's approach, with its Brouwerian motivation for accepting Nelson's classical formalism, would have suited Weyl best.