منابع مشابه
Lecture 19: Constructive Reals and Infinitesimal Calculus
We will present a very interesting theorem from Bishop and Bridges, and then we will briefly consider the nature of nonstandard models of the classical reals as presented in H.J. Keisler’s book, Elementary Calculus and his free 203 page instructor’s book entitled Foundations of the Infinitesimal Calculus [8] which is provided as a course resource with this lecture. We have not covered the basic...
متن کاملEquisingularity in R as Morse Stability in Infinitesimal Calculus
Two seemingly unrelated problems are intimately connected. The first is the equsingularity problem in R: For an analytic family ft : (R , 0) → (R, 0), when should it be called an “equisingular deformation”? This amounts to finding a suitable trivialization condition (as strong as possible) and, of course, a criterion. The second is on the Morse stability. We define R∗, which is R “enriched” wit...
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From today’s perspective, a mathematical technique that lacks rigor or leads to paradoxes is a contradiction in terms. It must be expelled from mathematics, lest it discredit the profession, sow chaos, and put a large stain on the shining surface of eternal truth. This is precisely what the leading mathematicians of the most learned Catholic order, the Jesuits, said about the “method of indivis...
متن کاملCompleteness of the Leibniz Field and Rigorousness of Infinitesimal Calculus
We present a characterization of the completeness of the field of real numbers in the form of a collection of ten equivalent state ments borrowed from algebra, real analysis, general topology and non-standard analysis. We also discuss the completeness of nonArchimedean fields and present several examples of such fields. As an application we exploit one of our results to argue that the Leibniz ...
متن کاملFrom Pythagoreans and Weierstrassians to True Infinitesimal Calculus
In teaching infinitesimal calculus we sought to present basic concepts like continuity and convergence by comparing and contrasting various definitions, rather than presenting “the definition” to the students as a monolithic absolute. We hope that our experiences could be useful to other instructors wishing to follow this method of instruction. A poll run at the conclusion of the course indicat...
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ژورنال
عنوان ژورنال: Nature
سال: 1892
ISSN: 0028-0836,1476-4687
DOI: 10.1038/046218a0