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The end-point method is mathematically developed and its application to the Milne kernel studied in detail. The general solution of the Wiener-Hopf integral equation is first obtained. The Mime kernel appears in applying this method to the integral equation describing the diffusion and multiplication of neutrons in multiplying and scattering media. The neutrons are treated as monochromatic, isotropically scattered and of the same total mean free path in all materials involved. Only problems with spherical symmetry are treated, these being reducible to equivalent infinite slab problems. Solutions are obtained for tamped and untamped spheres; in the former case both growing and decaying exponential asymptotic solutions in the tamper are treated in detail. Appendix I treats the effects of the approximations inherent in the end-point method (cf. LADC - 79). Appendix II gives the solution of the inhomogeneous Wiener-Hopf equation. (AN)
An analysis of Newton's mathematical work, from early discoveries to mature reflections, and a discussion of Newton's views on the role and nature of mathematics. Historians of mathematics have devoted considerable attention to Isaac Newton's work on algebra, series, fluxions, quadratures, and geometry. In Isaac Newton on Mathematical Certainty and Method, Niccolò Guicciardini examines a critical aspect of Newton's work that has not been tightly connected to Newton's actual practice: his philosophy of mathematics. Newton aimed to inject certainty into natural philosophy by deploying mathematical reasoning (titling his main work The Mathematical Principles of Natural Philosophy most probably to highlight a stark contrast to Descartes's Principles of Philosophy). To that end he paid concerted attention to method, particularly in relation to the issue of certainty, participating in contemporary debates on the subject and elaborating his own answers. Guicciardini shows how Newton carefully positioned himself against two giants in the “common” and “new” analysis, Descartes and Leibniz. Although his work was in many ways disconnected from the traditions of Greek geometry, Newton portrayed himself as antiquity's legitimate heir, thereby distancing himself from the moderns. Guicciardini reconstructs Newton's own method by extracting it from his concrete practice and not solely by examining his broader statements about such matters. He examines the full range of Newton's works, from his early treatises on series and fluxions to the late writings, which were produced in direct opposition to Leibniz. The complex interactions between Newton's understanding of method and his mathematical work then reveal themselves through Guicciardini's careful analysis of selected examples. Isaac Newton on Mathematical Certainty and Method uncovers what mathematics was for Newton, and what being a mathematician meant to him.
First published in 2004. Routledge is an imprint of Taylor & Francis, an informa company.
School mathematics is a complex subject and an ever-changing topic, but this book will help teachers, parents and employers to understand it better.
"Here, for the first time in English, are the philosophical essays - including the first statement of the "Duhem Thesis" - that formed the basis for Aim and Structure of Physical Theory, together with new translations of the historiographical essays presenting the equally celebrated "Continuity Thesis" by Pierre Duhem (1861-1916), a founding figure of the history and philosophy of science. Prefaced by an introduction on Duhem's intellectual development and continuing significance, here as well are important subsequent essays in which Duhem elaborated key concepts and critiqued such contemporaries as Henri Poincare and Ernst Mach. Together, these works offer a lively picture of the state of science at the turn of the century while addressing methodological issues that remain at the center of debate today."--BOOK JACKET.Title Summary field provided by Blackwell North America, Inc. All Rights Reserved