Fermat's dilemma: Why did he keep mum on infinitesimals? and the European theological context

TL;DR

This paper explores Fermat's use of adequality and infinitesimals within the 17th-century religious and philosophical context, revealing how theological debates influenced mathematical developments and interpretations.

Contribution

It provides a detailed analysis of Fermat's methods, clarifies their relation to infinitesimals, and examines the historical and theological factors affecting their interpretation.

Findings

Fermat's adequality relates to relations of infinite proximity.

Modern proxies help clarify Fermat's procedures.

Religious context influenced mathematical discourse of the era.

Abstract

The first half of the 17th century was a time of intellectual ferment when wars of natural philosophy were echoes of religious wars, as we illustrate by a case study of an apparently innocuous mathematical technique called adequality pioneered by the honorable judge Pierre de Fermat, its relation to indivisibles, as well as to other hocus-pocus. Andre Weil noted that simple applications of adequality involving polynomials can be treated purely algebraically but more general problems like the cycloid curve cannot be so treated and involve additional tools--leading the mathematician Fermat potentially into troubled waters. Breger attacks Tannery for tampering with Fermat's manuscript but it is Breger who tampers with Fermat's procedure by moving all terms to the left-hand side so as to accord better with Breger's own interpretation emphasizing the double root idea. We provide modern proxies for Fermat's procedures in terms of relations of infinite proximity as well as the standard part function. Keywords: adequality; atomism; cycloid; hylomorphism; indivisibles; infinitesimal; jesuat; jesuit; Edict of Nantes; Council of Trent 13.2