WHY INFINITESIMALS ARE NOT POPULAR

Modern Logic, 2001

Erkenntnis, 2013

Many historians of the calculus deny significant continuity between infinitesimal calculus of the 17th century and 20th century developments such as Robinson's theory. Robinson's hyperreals, while providing a consistent theory of infinitesimals, require the resources of modern logic; thus many commentators are comfortable denying a historical continuity. A notable exception is Robinson himself, whose identification with the Leibnizian tradition inspired Lakatos, Laugwitz, and others to consider the history of the infinitesimal in a more favorable light. Inspite of his Leibnizian sympathies, Robinson regards Berkeley's criticisms of the infinitesimal calculus as aptly demonstrating the inconsistency of reasoning with historical infinitesimal magnitudes. We argue that Robinson, among others, overestimates the force of Berkeley's criticisms, by underestimating the mathematical and philosophical resources available to Leibniz. Leibniz's infinitesimals are fictions, not logical fictions, as Ishiguro proposed, but rather pure fictions, like imaginaries, which are not eliminable by some syncategorematic paraphrase. We argue that Leibniz's defense of infinitesimals is more firmly grounded than Berkeley's criticism thereof. We show, moreover, that Leibniz's system for differential calculus was free of logical fallacies. Our argument strengthens the conception of modern infinitesimals as a development of Leibniz's strategy of relating inassignable to assignable quantities by means of his transcendental law of homogeneity.

Journal of Humanistic Mathematics

Felix Klein and Abraham Fraenkel each formulated a criterion for a theory of infinitesimals to be successful, in terms of the feasibility of implementation of the Mean Value Theorem. We explore the evolution of the idea over the past century, and the role of Abraham Robinson's framework therein.

Comments on the Difficulty and Validity of Various Approaches to the Calculus DAVIDTALL With the introduction of new infinitesimal methods in the last two decades, there are now available a number of dif ferent approaches to the calculus. In her perceptive review essay on "Infinitesimal Calculus" [3]. Peggi Marchi raised For the Learning of Mathematics 2, 2 (November 1981

infinitesimals are ininitely small quantities We may easily confuse the two concepts of infinitesimals , infinite divisible and infinite small (divisibles and idivisibles ,that they both claim to underlie modern calculus. this text is from my book "the invisible world of infinitesimals" with ISBN 978-3-8484-0002-7 in LAMBERT Academic publishing First, in mathematics, the concept of infinitesimal has appeared in geometric problems, as components of continuous magnitude, as opposed to that of the discrete number. Their first form is their intuitive Greek version of the Eudoxian method of exhaustion, referring to straight sections, areas, volumes, etc., without our basic concept of the limit.

Nonstandard Methods of Analysis, 1994

The most widely spread prejudice against infinitesimals resides in the opinion that the technique of infinitesimal analysis is extremely difficult to master. Moreover, it is usually emphasized that the nonstandard methods of analysis rest on rather sophisticated sections of set theory and mathematical logic. This circumstance is irrefutable but overrated, hampering in no way comprehension of infinitesimals. The purpose of this chapter is to corroborate the above statement by presenting the methodology of infinitesimal analysis at the routine level of rigor which is offered by the modern system of mathematical education invoking the naive set-theoretic stance that stems from Cantor. Alongside with elucidating the basic concepts of nonstandard set theory and its principles of transfer, idealization, and standardization, we pay attention also to comparing the new views of the basic concepts of analysis with those of the reverent inventors of the past. We hope so to witness the continual evolution and immortality of the ideas of differential and integral calculus which infinitesimal analysis in a today's disguise shed new light upon. 2.1. The Concept of Set in Infinitesimal Analysis In this section we will set forth a fragment of the foundations of infinitesimal analysis at the level of rigor close to the current practice of teaching calculus. 2.1.1. Contemporary courses in mathematical analysis rest usually on the concept of set. 2.1.2. Examples. (1) L. Schwartz, Analysis: "A set is a collection of objects. Examples: the set of all alumni of a school;

Antiquitates Mathematicae

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 Siméon-Denis Poisson, Gaspard-Gustave de Coriolis, and Jean-Nicolas Noël. We examine contrasting historiographic approaches to such lores, in the work of Laugwitz, Schubring, Spalt, and others, and address a recent critique by Archibald et al. We argue that the element of contingency in this history is more prominent than many modern historians seem willing to acknowledge.

Choice Reviews Online, 2014

We introduce a ring of the so-called Fermat reals, which is an extension of the real field containing nilpotent infinitesimals. The construction is inspired by Smooth Infini-tesimal Analysis (SIA) and provides a powerful theory of actual infinitesimals without any background in mathematical logic. In particular, in contrast to SIA, which admits models in intuitionistic logic only, the theory of Fermat reals is consistent with the classical logic. We face the problem of deciding whether or not a product of powers of nilpotent infinitesimals vanishes, study the identity principle for polynomials, and discuss the definition and properties of the total order relation. The construction is highly constructive, and every Fermat real admits a clear and order-preserving geometrical representation. Using nilpotent infinitesimals, every smooth function becomes a polynomial because the remainder in Taylor's formulas is now zero. Finally, we present several applications to informal classical calculations used in physics, and all these calculations now become rigorous, and at the same time, formally equal to the informal ones. In particular, an interesting rigorous deduction of the wave equation is given, which clarifies how to formalize the approximations tied with Hooke's law using the language of nilpotent infinitesimals.