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

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

Effects of rapid urbanisation on the urban thermal environment between 1990 and 2011 in Dhaka Megacity, Bangladesh

This study investigates the influence of land-use/land-cover (LULC) change on land surface temperature (LST) in Dhaka Megacity, Bangladesh during a period of rapid urbanisation. LST was derived from Landsat 5 TM scenes captured in 1990, 2000 and 2011 and compared to contemporaneous LULC maps. We compared index-based and linear spectral mixture analysis (LSMA) techniques for modelling LST. LSMA derived biophysical parameters corresponded more strongly to LST than those produced using index-based parameters. Results indicated that vegetation and water surfaces had relatively stable LST but it increased by around 2 °C when these surfaces were converted to built-up areas with extensive impervious surfaces. Knowledge of the expected change in LST when one land-cover is converted to another can inform land planners of the potential impact of future changes and urges the development of better management strategies

Phase transitional studies of polycrystalline Pb0.4Sr0.6TiO3Pb0.4Sr0.6TiO3 films using Raman scattering

Pb0.4Sr0.6TiO3Pb0.4Sr0.6TiO3 films with submicron size grains have been prepared on Pt substrates by the metalorganic decomposition method. X-ray diffraction analysis reveals that the films are polycrystalline with a perovskite crystal structure and negligible tetragonal splitting at room temperature. The room temperature Raman spectrum, however, shows all the characteristic phonon modes of a tetragonal ferroelectric phase. These modes lose their intensity with an increase in temperature but persist even up to 150 °C. This is in agreement with the dielectric permittivity versus temperature data which show a broad peak–room temperature, and ferroelectric hysteresis persisting up to ∼140 °C. Both Raman and dielectric data are interpreted as due to the presence of a distribution of phases in the film perhaps caused by a variation in Pb content along the film thickness and/or near grain boundary regions. © 2003 American Institute of Physics.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/70073/2/JAPIAU-93-3-1731-1.pd

The place of experimental tasks in geometry teaching: learning from the textbook designs of the early 20th century

The dual nature of geometry, in that it is a theoretical domain and an area of practical experience, presents mathematics teachers with opportunities and dilemmas. Opportunities exist to link theory with the everyday knowledge of pupils but the dilemmas are that learners very often find the dual nature of geometry a chasm that is very difficult to bridge. With research continuing to focus on understanding the nature of this problem, with a view to developing better pedagogical techniques, this paper examines the place of experimental tasks in the process of learning geometry. In particular, the paper provides some results from an analysis of innovative geometry textbooks designed in the early part of the 20th Century, a time when significant efforts were being made to improve the teaching and learning of geometry. The analysis suggests that experimental tasks have a vital role to play and that a potent tool for informing the design of such tasks, so that they build effectively on pupils’ geometrical intuition, is the notion of the geometrical eye, a term coined by Charles Godfrey in 1910 as “the power of seeing geometrical properties detach themselves from a figure"