A Burgessian critique of nominalistic tendencies in contemporary mathematics and its historiography

We analyze the developments in mathematical rigor from the viewpoint of a Burgessian critique of nominalistic reconstructions. We apply such a critique to the reconstruction of infinitesimal analysis accomplished through the efforts of Cantor, Dedekind, and Weierstrass; to the reconstruction of Cauchy's foundational work associated with the work of Boyer and Grabiner; and to Bishop's constructivist reconstruction of classical analysis. We examine the effects of a nominalist disposition on historiography, teaching, and research.Comment: 57 pages; 3 figures. Corrected misprint

Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond

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.Comment: 69 pages, 3 figure

Ten Misconceptions from the History of Analysis and Their Debunking

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 oversimplification created by triumvirate followers, who tend to view the history of analysis as a pre-ordained march toward the radiant future of Weierstrassian epsilontics. In the present text, we document distortions of the history of analysis stemming from the triumvirate ideology of ontological minimalism, which identified the continuum with a single number system. Such anachronistic distortions characterize the received interpretation of Stevin, Leibniz, d'Alembert, Cauchy, and others.Comment: 46 pages, 4 figures; Foundations of Science (2012). arXiv admin note: text overlap with arXiv:1108.2885 and arXiv:1110.545

Variational Analysis Down Under Open Problem Session

© 2018, Springer Science+Business Media, LLC, part of Springer Nature. We state the problems discussed in the open problem session at Variational Analysis Down Under conference held in honour of Prof. Asen Dontchev on 19–21 February 2018 at Federation University Australia

Understanding (in) Newton’s Argument for Universal Gravitation.

In this essay, I attempt to assess Henk De Regt and Dennis Dieks recent pragmatic and contextual account of scientific understanding on the basis of an important historical case-study: understanding in Newton’s theory of universal gravitation and Huygens’ reception of universal gravitation. It will be shown that de Regt and Dieks’ CIT-criterion, which stipulates that the appropriate combination of scientists’ skills and intelligibility-enhancing theoretical virtues is a condition for scientific understanding, is too strong. On the basis of this case-study, it will be shown that scientists can understand each others’ positions qualitatively and quantitatively, despite their endorsement of different worldviews and despite their convictions as what counts as a proper explanation

RANKINE: a Computer Software Package for the Analysis and Design of Steam Power Generating Units

Introduction Nearly all major electric-generating power systems run through a series of processes which is based upon the theoretical thermodynamic cycle called the Rankine cycle. Because the Rankine cycle is the basis for steam power cycles, its analysis is vital to the design of power systems and to ensuring their proper operation. By introducing additional components, such as feedwater heaters or reheat legs, or adjusting the operating temperatures and pressures, a Rankine cycle analysis can be used to maximize the thermal efficiency and exergetic efficiency (second law effectiveness) within the constraints of turbine exit quality and pump capacity. The process of analyzing these Rankine cycles and attempts at optimizing the operating conditions can be very complex as well as time consuming. Such an analysis dictates the use of the computer. A software package (RANKINE) has been designed and tested which will solve a wide range of Rankine cycle problems very quickly. This software package is the subject of this paper. Another use of the software package is in the instruction of undergraduate mechanical engineering students. Because the Rankine cycle is the basis for steam power cycles, it is a primary topic covered in an intermediate thermodynamics course. In order to provide the students with a feeling for how the ideal Rankine cycle may be modified in order to model any actual steam power cycle, a large number of examples need to be worked. With the RANKINE software package, the students will be able to gain a feeling for actual steam power cycles in a reasonable time period. Previous software packages dealing with Rankine or modified Rankine cycles fall into one of three groups. There have been some very simple programs developed which can b