3,382 research outputs found
Fermat, Leibniz, Euler, and the gang: The true history of the concepts of limit and shadow
Fermat, Leibniz, Euler, and Cauchy all used one or another form of
approximate equality, or the idea of discarding "negligible" terms, so as to
obtain a correct analytic answer. Their inferential moves find suitable proxies
in the context of modern theories of infinitesimals, and specifically the
concept of shadow. We give an application to decreasing rearrangements of real
functions.Comment: 35 pages, 2 figures, to appear in Notices of the American
Mathematical Society 61 (2014), no.
When is .999... less than 1?
We examine alternative interpretations of the symbol described as nought,
point, nine recurring. Is "an infinite number of 9s" merely a figure of speech?
How are such alternative interpretations related to infinite cardinalities? How
are they expressed in Lightstone's "semicolon" notation? Is it possible to
choose a canonical alternative interpretation? Should unital evaluation of the
symbol .999 . . . be inculcated in a pre-limit teaching environment? The
problem of the unital evaluation is hereby examined from the pre-R, pre-lim
viewpoint of the student.Comment: 28 page
Cauchy's infinitesimals, his sum theorem, and foundational paradigms
Cauchy's sum theorem is a prototype of what is today a basic result on the
convergence of a series of functions in undergraduate analysis. We seek to
interpret Cauchy's proof, and discuss the related epistemological questions
involved in comparing distinct interpretive paradigms. Cauchy's proof is often
interpreted in the modern framework of a Weierstrassian paradigm. We analyze
Cauchy's proof closely and show that it finds closer proxies in a different
modern framework.
Keywords: Cauchy's infinitesimal; sum theorem; quantifier alternation;
uniform convergence; foundational paradigms.Comment: 42 pages; to appear in Foundations of Scienc
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