Topic
Standard part function
About: Standard part function is a research topic. Over the lifetime, 12 publications have been published within this topic receiving 70 citations.
Papers
TL;DR: The first half of the 17th century was a time of intellectual ferment when wars of natural philosophy were echoes of religious wars, as illustrated by a case study of an apparently innocuous mathematical technique called adequality pioneered by the honorable judge Pierre de Fermat, its relation to indivisibles, as well as to other hocus-pocus as discussed by the authors.
Abstract: The first half of the 17th century was a time of intellectual ferment when wars of natural philosophy were echoes of religious wars, as we illustrate by a case study of an apparently innocuous mathematical technique called adequality pioneered by the honorable judge Pierre de Fermat, its relation to indivisibles, as well as to other hocus-pocus. Andre Weil noted that simple applications of adequality involving polynomials can be treated purely algebraically but more general problems like the cycloid curve cannot be so treated and involve additional tools--leading the mathematician Fermat potentially into troubled waters. Breger attacks Tannery for tampering with Fermat's manuscript but it is Breger who tampers with Fermat's procedure by moving all terms to the left-hand side so as to accord better with Breger's own interpretation emphasizing the double root idea. We provide modern proxies for Fermat's procedures in terms of relations of infinite proximity as well as the standard part function.
Keywords: adequality; atomism; cycloid; hylomorphism; indivisibles; infinitesimal; jesuat; jesuit; Edict of Nantes; Council of Trent 13.2
7 citations
TL;DR: A direct and elementary proof of the fact that every real- valued probability measure can be approximated up to an infinitesimal by a hyperreal-valued one which is regular and defined on the whole powerset of the sample space is given.
Abstract: We give a direct and elementary proof of the fact that every real- valued probability measure can be approximated up to an infinitesimal by a hyperreal-valued one which is regular and defined on the whole powerset of the sample space.
5 citations
Posted Content•
TL;DR: In this article, the ambiguity of an ellipsis is modeled as a generic limit of B. Cornu and D. Tall, and the student resistance to the unital evaluation of.999... is directed against an unspoken and unacknowledged application of the standard part function, namely the stripping away of a ghost of an infinitesimal, to echo George Berkeley.
Abstract: Is .999... equal to 1? Lightstone's decimal expansions yield an infinity of numbers in [0,1] whose expansion starts with an unbounded number of digits "9". We present some non-standard thoughts on the ambiguity of an ellipsis, modeling the cognitive concept of generic limit of B. Cornu and D. Tall. A choice of a non-standard hyperinteger H specifies an H-infinite extended decimal string of 9s, corresponding to an infinitesimally diminished hyperreal value. In our model, the student resistance to the unital evaluation of .999... is directed against an unspoken and unacknowledged application of the standard part function, namely the stripping away of a ghost of an infinitesimal, to echo George Berkeley. So long as the number system has not been specified, the students' hunch that .999... can fall infinitesimally short of 1, can be justified in a mathematically rigorous fashion.
4 citations
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TL;DR: The classical ERNA (elementary recursive nonstandard analysis) as mentioned in this paper is a constructive system of non-standard analysis proposed around 1995 by Chuaqui, Suppes and Sommer.
Abstract: Elementary Recursive Nonstandard Analysis, in short ERNA, is a constructive system of nonstandard analysis proposed around 1995 by Chuaqui, Suppes and Sommer. It has been shown to be consistent and, without standard part function or continuum, it allows major parts of analysis to be developed in an applicable form. We briefly discuss ERNA’s foundations and use them to prove a supremum principle and provide a square root function, both up to infinitesimals.
3 citations
10 Oct 2015
TL;DR: The field of Omicran-reals O as mentioned in this paper is an extension of real numbers which contains the infinite and infinitesimal numbers as the field of hyperreals.
Abstract: From the works of Abraham Robinson, we know that the heuristic idea of infinite and infinitesimal numbers has obtained a formal rigor, he proved that the field of real numbers R can be considered as a proper subset of a new field, * R, which is called field of hyperreal [1] numbers and contains the infinite and infinitesimal numbers From the approach of Robinson we can represent every hyperreal by a sequence o f R N modulo a maximal ideal M, this ideal is defined by using an ultrafilter U Unfortunately, the Ultrafilter U and the order relation defined on * R are unknown, only the existence can be proved by the axiom of choice In this paper, we find a new extension of real numbers which contains the infinite and infinitesimal numbers as the field of hyperreals, the new set is called the field of Omicran-reals O Moreover, the new approach of construction is simple compared to other methods [1, 7, 8]and very precise, and the field O is endowed with a well defined total order relation
2 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2018 | 2 |
| 2017 | 1 |
| 2016 | 1 |
| 2015 | 1 |
| 2012 | 2 |
| 2008 | 1 |