An Examination of the Historical Trajectory of Infinitesimals
The paper "Episodes from the History of Infinitesimals" by Mikhail G. Katz offers a comprehensive analysis of the philosophical, mathematical, and theological dimensions of infinitesimals from the post-Leibnizian era onwards. This investigation into the history of infinitesimals underscores the evolving perceptions and uses of these abstract mathematical entities, marked by the contrasting views of significant historical figures.
The paper opens with a discussion on the critical stance towards infinitesimals in the 18th century, notably through d'Alembert's description of them as "chimeras." This derogatory term persisted into the 19th and 20th centuries, with important figures like Abbé Moigno and Alain Connes echoing this sentiment. The opposition largely viewed infinitesimals as illusory, with metaphysical and logical objections underpinning their arguments.
Despite such opposition, infinitesimals were embedded in the work of eminent mathematicians like Cauchy, Poisson, and Riemann well into the 19th century. Cauchy's distinction between "epsilontic" and infinitesimal epsilons exemplifies his nuanced use of infinitesimals in calculus and analysis. Riemann extended these ideas to geometry, impacting subsequent thinkers like Levi-Civita. These scholarly activities exhibit the practical utility of infinitesimals in mathematical reasoning.
Cantor's late 19th-century critiques of infinitesimals marked a significant pivot. His insistence on formal structures that exclude non-Cantorian forms reflects broader trends toward rigor and formalism, which characterize much of the mathematics of that era. Cantor's dismissal resided in the belief that any theory incapable of consistent operations like multiplication by transfinite numbers lacked validity.
Contrastingly, Peano's gradual acceptance and formalization of infinitesimals indicate a nascent revival. Peano's work, among others, preluded the later resurgence of infinitesimal analysis, leading to developments by scholars like Skolem, Hewitt, and Robinson. Robinson's 20th-century non-standard analysis reestablished infinitesimals within mathematics, offering a formal grounding that addressed many historical criticisms.
The implications of this historical trajectory are multifaceted. Practically, the ongoing dialogue highlights the persistent relevance of infinitesimals across various domains of mathematics, including real analysis and differential geometry. Theoretically, it prompts reflections on the nature of mathematical foundations and the philosophical underpinnings of mathematical thought.
This paper not only chronicles the voyages of infinitesimals but also sheds light on the broader landscape of mathematical evolution, encouraging a reevaluation of mathematical orthodoxies and inviting future exploration into abstract mathematical concepts like infinitesimals within new frameworks. As such, it makes a valuable contribution to the historical and philosophical discourse surrounding one of mathematics' most enduring and controversial ideas.