Infinitesimals as an issue in neo-Kantian philosophy of science

TL;DR

This paper explores how the Marburg neo-Kantian philosophy of science critically engaged with the mathematical revolution involving infinitesimals, emphasizing their nuanced philosophical stance amidst foundational debates.

Contribution

It reveals the Marburg school's sophisticated philosophical approach to limits and infinitesimals, contrasting with both traditional and modern orthodoxies.

Findings

Marburg neo-Kantianism offered a nuanced view on infinitesimals.

The school aimed to clarify Leibniz's principle of continuity.

They sought to reconcile mathematical rigor with philosophical coherence.

Abstract

We seek to elucidate the philosophical context in which one of the most important conceptual transformations of modern mathematics took place, namely the so-called revolution in rigor in infinitesimal calculus and mathematical analysis. Some of the protagonists of the said revolution were Cauchy, Cantor, Dedekind, and Weierstrass. The dominant current of philosophy in Germany at the time was neo-Kantianism. Among its various currents, the Marburg school (Cohen, Natorp, Cassirer, and others) was the one most interested in matters scientific and mathematical. Our main thesis is that Marburg neo-Kantian philosophy formulated a sophisticated position towards the problems raised by the concepts of limits and infinitesimals. The Marburg school neither clung to the traditional approach of logically and metaphysically dubious infinitesimals, nor whiggishly subscribed to the new orthodoxy of the "great triumvirate" of Cantor, Dedekind, and Weierstrass that declared infinitesimals conceptus nongrati in mathematical discourse. Rather, following Cohen's lead, the Marburg philosophers sought to clarify Leibniz's principle of continuity, and to exploit it in making sense of infinitesimals and related concepts.