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Results tagged with nonstandard-analysis
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Nonstandard analysis is a way of doing calculus and analysis with infinitesimals. The historical approach of Leibniz, Euler, and others to infinitesimal calculus was gradually replaced by epsilon, delta techniques in the context of a real continuum, in the 19th century. It was not until the 1960s that Abraham Robinson developed a theory of a hyperreal continuum that allows for a development of analysis procedurally akin to that of its founders.
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What are the advantages of the more abstract approaches to nonstandard analysis?
This question does not concern the comparative merits of standard (SA) and nonstandard (NSA) analysis but rather a comparison of different approaches to NSA. What are the concrete advantages of the ab …
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What is... a grossone?
Y. Sergeyev developed a positional system for representing infinite numbers using a basic unit called a "grossone", as well as what he calls an "infinity computer". The mathematical value of this see …
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Euler's mathematics in terms of modern theories?
Some aspects of Euler's work were formalized in terms of modern infinitesimal theories by Laugwitz, McKinzie, Tuckey, and others. Referring to the latter, G. Ferraro claims that "one can see in operat …
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Was the early calculus inconsistent?
This question does NOT concern the RIGOR, or lack thereof, of the early calculus. Rather the question is of its CONSISTENCY.
George Berkeley wrote in 1734 with reference to the early calculus that s …
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Did Lagrange change his mind about infinitesimals?
Lagrange is famous for his attempt to found analysis algebraically using power series expansions, an approach that, as we know today, is limited to analytic functions. Lagrange is also known as the in …
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Bibliographic request concerning an article by Bernstein and Robinson
Concerning the article "Bernstein, Allen R.; Robinson, Abraham.
Solution of an invariant subspace problem of K. T. Smith and
P. R. Halmos. Pacific J. Math. 16 1966 421-431" I am interested in
finding …
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9
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answers
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differential geometry using Robinson's infinitesimals?
Is there a detailed treatment of differential geometry using Robinson's infinitesimals?
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What's Reeb's take on naive integers?
Georges Reeb's "claim Q" is the statement that "naive integers don't fill up $\mathbb{N}$". To anyone familiar with model theory this could easily be interpreted as the existence of nonstandard models …
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Which universities teach true infinitesimal calculus? [closed]
My colleague and I are currently teaching "true infinitesimal calculus" (TIC), in the sense of calculus with infinitesimals, to a class of about 120 freshmen at our university, based on the book by Ke …
8
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1
answer
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Who said "the naive counting numbers don't exhaust $\Bbb N$"?
In the context of Robinson's framework, or more precisely its reformulation by Ed Nelson, one of the practitioners in the field expressed the sentiment something like "the naive counting numbers don't …
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Salvaging Leibnizian formalism?
Can one justify Leibniz's formalism in a suitable algebraic or topological context?
We have published some papers recently where we argue that Leibniz's formalism for the calculus wasn't inconsisten …
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Was Cauchy prescient?
Cauchy proved a sum theorem for series of continuous functions in 1821, and published another article on the subject in 1853.
Michael Segre, writing in Archive for History of Exact Sciences, claimed …
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Is there a source linking Robinson's work in wing theory with his theory of infinitesimals?
Abraham Robinson worked in applied mathematics for several decades. MathSciNet lists 12 articles by Robinson in wing theory. His production included the book
Robinson, A.; Laurmann, J. A. Wing theory …
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Did Bishop make those comments in his oral presentation?
The 1975 published version of a 1974 talk at a workshop by Errett Bishop contains the following comment:
"A more recent attempt at mathematics by formal finesse is non-standard analysis. I gather …
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Standard models of N and R: An Alice/Bob approach
This is a question about a comment in a recent publication by Roman
Kossak. Kossak wrote:
"Nonstandardness in set theory has a different nature. In
arithmetic, there is one intended object of study …
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