相关论文: Episodes from the history of infinitesimals

A refinement of the classic equivalence relation among Cauchy sequences yields a useful infinitesimal-enriched number system. Such an approach can be seen as formalizing Cauchy's sentiment that a null sequence "becomes" an infinitesimal. We…

In the history of infinitesimal calculus, we trace innovation from Leibniz to Cauchy and reaction from Berkeley to Mansion and beyond. We explore 19th century infinitesimal lores, including the approaches of Simeon-Denis Poisson,…

We examine prevailing philosophical and historical views about the origin of infinitesimal mathematics in light of modern infinitesimal theories, and show the works of Fermat, Leibniz, Euler, Cauchy and other giants of infinitesimal…

19th century real analysis received a major impetus from Cauchy's work. Cauchy mentions variable quantities, limits, and infinitesimals, but the meaning he attached to these terms is not identical to their modern meaning. Some Cauchy…

Infinitesimals are natural products of the human imagination. Their history goes back to the Greek antiquity. Their role in the calculus and analysis has seen dramatic ups and downs. They have stimulated strong opinions and even vitriol.…

We explore the issue of providing a foundational framework for Leibnizian infinitesimals in the light of modern standard and nonstandard approaches. We outline a trichotomy of ordinals, cardinals and ringinals as a historiographic tool. A…

This is a brief overview of some turning points in the history of infinitesimals.

Fermat, Leibniz, Euler, and Cauchy all used one or another form of approximate equality, or the idea of discarding "negligible" terms, so as to obtain a correct analytic answer. Their inferential moves find suitable proxies in the context…

This is a survey of several approaches to the framework for working with infinitesimals and infinite numbers, originally developed by Abraham Robinson in the 1960s, and their constructive engagement with the Cantor-Dedekind postulate and…

We show that the field of complex numbers $\mathbb C$ contains non-zero infinitesimals by observing that $\mathbb C$ contains non-Archimedean subfields. Our observation is based on an old theorem in algebra due to E. Steinitz, discussed in…

Philosopher Benardete challenged both the conventional wisdom and the received mathematical treatment of zero, dot, nine recurring. An initially puzzling passage in Benardete on the intelligibility of the continuum reveals challenging…

We contribute to the lively debate in current scholarship on the Leibnizian calculus. In a recent text, Arthur and Rabouin argue that non-Archimedean continua are incompatible with Leibniz's concepts of number, quantity and magnitude. They…

The widespread idea that infinitesimals were "eliminated" by the "great triumvirate" of Cantor, Dedekind, and Weierstrass is refuted by an uninterrupted chain of work on infinitesimal-enriched number systems. The elimination claim is an…

Abraham Robinson's framework for modern infinitesimals was developed half a century ago. It enables a re-evaluation of the procedures of the pioneers of mathematical analysis. Their procedures have been often viewed through the lens of the…

Mathematical concepts and results have often been given a long history, stretching far back in time. Yet recent work in the history of mathematics has tended to focus on local topics, over a short term-scale, and on the study of ephemeral…

In a previous article we gave the general foundations of the theory of movement considered from a philosophical and mathematical point of view. Philosophical it meant to understand the opposition of the one and the multiple, mathematically…

Like his colleagues de Prony, Petit, and Poisson at the Ecole Polytechnique, Cauchy used infinitesimals in the Leibniz-Euler tradition both in his research and teaching. Cauchy applied infinitesimals in an 1826 work in differential geometry…

Leibniz used the term fiction in conjunction with infinitesimals. What kind of fictions they were exactly is a subject of scholarly dispute. The position of Bos and Mancosu contrasts with that of Ishiguro and Arthur. Leibniz's own views,…

The presence of infinitesimals is traced back to some of the most general algebraic structures, namely, semigroups, and in fact, magmas, [1], in which none of the structures of linear order, field, or the Archimedean property need to be…

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…