Use Of Infinitesimals Research Articles

The evil epsilon or it is not reasonable to be a heretic mathematician

It was clear to Berkeley and Robinson that the use of infinitesimals as a process is wrong and poses logical and epistemological problems, making them impossible to be perceived. Both of them believed that ‘existing’ means ‘capable of being grasped.' In spite of their criticism of referring to infinitesimals as a process, their solution to the problem was completely different. Berkeley sacrificed mathematics and science on the altar of epistemology and theology, while Robinson refused to give up anything; he wanted it all: epistemology, mathematics and theology, a goal which he achieved using rigorous logic in his construction of non-standard analysis.

19th-century real analysis, forward and backward

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 historians work in a conceptual scheme dominated by an assumption of a teleological nature of the evolution of real analysis toward a preordained outcome. Thus, Gilain and Siegmund-Schultze assume that references to limite in Cauchy's work necessarily imply that Cauchy was working with an Archi-medean continuum, whereas infinitesimals were merely a convenient figure of speech, for which Cauchy had in mind a complete justification in terms of Archimedean limits. However, there is another formalisation of Cauchy's procedures exploiting his limite, more consistent with Cauchy's ubiquitous use of infinitesimals, in terms of the standard part principle of modern infinitesimal analysis. We challenge a misconception according to which Cauchy was allegedly forced to teach infinitesimals at the Ecole Polytechnique. We show that the debate there concerned mainly the issue of rigor, a separate one from infinitesimals. A critique of Cauchy's approach by his contemporary de Prony sheds light on the meaning of rigor to Cauchy and his contemporaries. An attentive reading of Cauchy's work challenges received views on Cauchy's role in the history of analysis, and indicates that he was a pioneer of infinitesimal techniques as much as a harbinger of the Epsilontik.

Margins of authority: Coke's Institutes and the epistemology of the string cite

ABSTRACTPrecedential authority has an important place in doctrinal explication and analysis in Anglo-American law. Efforts to manage these precedents visually, in sixteenth- and seventeenth-century England, display a variety of solutions to the question of where precedents belong and how to represent them. Sir Edward Coke’s approach, in his Institutes (1628–44), was to place the citations in the margin; this way of managing textual authority connected legal concepts in a way that lent new significance to the citations. The discussion examines this development by considering it in relation to the introduction of perspective in visual art, in the early fifteenth century, and the use of infinitesimals in the new mathematics of the early seventeenth century.

Margins of Authority: Coke's Institutes and the Epistemology of the String Cite

Precedential authority has an important place in doctrinal explication and analysis in Anglo-American law. Efforts to manage these precedents visually, in sixteenth- and seventeenth-century England, display a variety of solutions to the question of where precedents belong and how to represent them. Sir Edward Coke’s approach, in his Institutes (1628-44), was to place the citations in the margin; this way of managing textual authority connected legal concepts in a way that lent new significance to the citations. The discussion examines this development by considering it in relation to the introduction of perspective in visual art, in the early fifteenth century, and the use of infinitesimals in the new mathematics of the early seventeenth century.

Controversies in the Foundations of Analysis: Comments on Schubring’s Conflicts

Foundations of Science recently published a rebuttal to a portion of our essay it published two years ago. The author, G. Schubring, argues that our 2013 text treated unfairly his 2005 book, Conflicts between generalization, rigor, and intuition. He further argues that our attempt to show that Cauchy is part of a long infinitesimalist tradition confuses text with context and thereby misunderstands the significance of Cauchy's use of infinitesimals. Here we defend our original analysis of variousmisconceptions and misinterpretations concerning the history of infinitesimals and, in particular, the role of infinitesimals in Cauchy's mathematics. We show that Schubring misinterprets Proclus, Leibniz, and Klein on non-Archimedean issues, ignores the Jesuit context of Moigno's flawed critique of infinitesimals, and misrepresents, to the point of caricature, the pioneering Cauchy scholarship of D. Laugwitz. Keywords: Archimedean axiom, Cauchy, Felix Klein, hornangle, infinitesimal, Leibniz, ontology, procedure

On a nonstandard Brownian motion and its maximal function

This article uses Radically Elementary Probability Theory (REPT) to prove results about the Wiener walk (the radically elementary Brownian motion) without the technical apparatus required by stochastic integration. The techniques used replace measure-theoretic tools by discrete probability and the rigorous use of infinitesimals. Specifically, REPT is applied to the results in Palacios (The American Statistician, 2008) to calculate certain expectations related to the Wiener walk and its maximal function. Because Palacios uses mostly combinatorics and no measure theory his results carry over through REPT with minimal changes. The paper also presents a construction of the Wiener walk which is intended to mimic the construction of Brownian motion from “continuous” white noise. A brief review of the nonstandard model on which REPT is based is given in the Appendix in order to minimize the need for previous exposure to the subject.

A história da derivada de Mariana: uma experiência didática

Neste artigo introduzimos o conceito de derivada como quociente de infinitésimos através de uma história para crianças e adultos. Incluímos o relato da compreensão, da história e do conceito, demonstradas por duas meninas, uma de 10, outra de 12 anos. Segue uma breve apologia em favor do uso dos infinitésimos no ensino de cálculo e com a descrição do contexto de uma sala de aula de cálculo para calouros, onde essa história foi aplicada como ficha de trabalho. Lançamos um repto aos professor de cálculo e autores de livros textos: produzam algo igualmente acessível para apresentar o conceito de derivada pela via dos limites.

Free-variable axiomatic foundations of infinitesimal analysis: A fragment with finitary consistency proof

In treatises or advanced textbooks on theoretical physics, it is apparent that the way mathematics is used is very different from what is to be found in books of mathematics. There is, for example, no close connection between books on analysis, on the one hand, and any classical textbook in quantum mechanics, for example, Schiff, [11], or quite recent books, for example Ryder, [10], on quantum field theory. The differences run a good deal deeper than the fact that the books on theoretical physics are not written in the definition-theorem-proof style characteristic of pure mathematics. Although a good many propositions are proved in the books on physics, there are almost with exception no existential proofs, and consequently there is no really serious systematic use of quantifiers. Another important characteristic is the free use of infinitesimals. In fact, most results would not lose anything, from a physicist's point of view, by leaving them in approximate form, i.e., instead of strict equalities or inequalities, using equalities or inequalities only up to an infinitesimal.The discrepancy between the way mathematics is ordinarily done in theoretical physics and the way it is built up from a foundational standpoint in any of the standard modern views raises the question of whether it might be possible to construct quite directly a rigorous foundation that reflects a significant part of this standard practice in theoretical physics. Other parts of standard practice in physics, for example, the use of physically intuitive but nonrigorous arguments, are not present in our system.

About electrons crossing nodes

Author defends a critique of an article published in August 1990 in this Journal by explaining how the controversial difference in understanding and use of infinitesimals causes the discrepancy bei...

Early delta functions and the use of infinitesimals in research

RÉSUMÉ. — Vers 1820, dans leurs travaux de physique mathématique, Cauchy, Fourier et Poisson utilisaient des méthodes qui seront redécouvertes plus d'un siècle plus tard. Les fonctions delta et les méthodes de sommation des séries divergentes et des intégrales étaient fondées sur l'emploi de quantités infiniment petites et infiniment grandes qui étaient développées de façon systématique dans les manuels de Cauchy.

Note on the Sphero-Conic

(i.) A quadric cone is a cone having any conic for base, and a point O on the line perpendicular to the plane of the conic through the focus for vertex.(ii.) A sphero-conic is the curve in which any quadric cone cuts a sphere whose centre is the vertex of the cone.In an introductory course on space-geometry, where only a passing mention of sphero-conics seems possible, I have found it useful to show directly the fundamental property that the sum or difference of the focal distances is constant, instead of following the classical method, which consists in deducing the property by the use of infinitesimals, from the fact that these focal distances make equal angles with the curve; this latter fact being in effect a direct consequence of Desargue’s theorem.

Elements of the Differential and Integral Calculus, by a new Method, founded on the True System of Sir Isaac Newton, without the Use of Infinitesimals or Limits Elements of the Infinitesimal Calculus, with Numerous Examples and Applications to Analysis and Geometry On a New Method of Obtaining the Differentials of Functions with Especial Reference to the Newtonian Conception of Rates

AN American writer who had exceptional opportunities of contrasting the methods of mathematical teaching adopted in his own country with those which obtained at Cambridge twenty-five years ago, strongly condemns the Transatlantic system, and leads his readers to infer that the attainments of the ordinary graduate in this particular branch of study were only on a par with those of a fairly trained schoolboy here. It may be supposed, then, that not many of the students ventured upon the difficulties of the calculus. Indeed, he writes that “at Yale where the course used to be thought a very difficult and thorough one, the Differential was among the optional studies at the end of the third year.” (Bristed: “Five Years at an English University,” vol. ii., pp. 94, &c., 1852.) Elements of the Differential and Integral Calculus, by a new Method, founded on the True System of Sir Isaac Newton, without the Use of Infinitesimals or Limits. By C. P. Buckingham. (Chicago: S. C. Griggs and Co., 1875. 343 pp.) Elements of the Infinitesimal Calculus, with Numerous Examples and Applications to Analysis and Geometry. By James G. Clark (Ray Series. New York: Wilson, Hinkle, and Co., 1875. 441 pp.) On a New Method of Obtaining the Differentials of Functions with Especial Reference to the Newtonian Conception of Rates or Velocities. By J. Minot Rice W. Woolsey Johnson. (New York: D. van Nosbrand, 1875. 32 pp.)

Exposition Géométrique des propriétés générales des Courbes Eléments de Calcul approximatif

WE have read these works with interest and somewhat of surprise: with interest because the subjects are fairly interesting and are treated in the well-marked style which distinguishes the writings of French mathematicians; with somewhat of surprise that the subjects treated at such length should have met with such a large circle of readers as is indicated by the number of editions that have been called for. The first work on our list establishes many general properties of curves by means of first principles and by the use of infinitesimals. This mode of treatment, so far as we know, is confined in our own text-books to a chapter or two in Dr. Salmonapos;s works, and it would be hard to find more than he has given in any other work. The author himself states that this elementary knowledge will carry the student through the book with the sole exception that a more extended acquaintance with mathematics is required for an article devoted to the finding the distance between a curve and its osculating sphere in the neighbourhood of the point of contact. The author, too, claims the major part of the demonstrations as his own, though in some cases he has generalised results previously given, and in some cases has established known properties in a novel way.