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Results tagged with nonstandard-analysis
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user 28128
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.
7
votes
Did Bishop make those comments in his oral presentation?
We were able to obtain an audio file of Bishop's talk from the American Academy of Arts and Sciences. Our analysis will be published in Historia Mathematica and is available on the arxiv.
We exami …
- 16.6k
1
vote
Lapses of "the early proponents of the doctrine of limits"
This is mainly a comment on the discussion following the original post. Leibniz did not use the $\Sigma^\infty$ notation. If you replace $\infty$ by an infinite hyperinteger, and interpret the sum a …
- 16.6k
1
vote
Which universities teach true infinitesimal calculus?
So far I have been able to find out that true infinitesimal calculus was taught at the following universities:
University of Hawaii;
University of Illinois at Urbana-Champaign;
University of Iowa;
…
1
vote
Literature that helps explain what the theory of numerosities contributes with
The theory of numerosities is more consistent with pre-mathematical intuitions of how collections (not to use the technical term set) should behave relative to each other. The fact that such an altern …
- 16.6k
1
vote
Can there be a numerical system in which logarithms can be expressed in terms of exponential...
As you suggested at a related site, the natural log can be expressed via the shadow of a hyperfinite partial sum of the harmonic series.
- 16.6k
0
votes
Accepted
Nonstandard definition for the generator of a standard Ito diffusion
try F. Herzberg at http://link.springer.com/chapter/10.1007/978-3-642-33149-7_7
- 16.6k
6
votes
differential geometry using Robinson's infinitesimals?
Somewhat belatedly we developed foundations for differential geometry using infinitesimal displacements here:
Nowik, T.; Katz, M. "Differential geometry via infinitesimal displacements." Journal of L …
- 16.6k
5
votes
Is non-existence of the hyperreals consistent with ZF?
In response specifically to the title of the question: "Is non-existence of the hyperreals consistent with ZF?", technically speaking the answer is NO. Kanovei and Shelah constructed a definable mode …
- 16.6k
2
votes
How helpful is non-standard analysis?
Steve Huntsman's claim attributed to wikipedia that "the list of new applications in mathematics is still very small" is patently false. In fact, I was unable to find such a claim there. To mention ju …
- 16.6k
0
votes
Would Euler's proofs get published in a modern math Journal, especially considering his trea...
In answering this question, it is helpful to make a distinction between, on the one hand, what Reeder calls the "inferential moves" that Euler makes (see related thread Euler's mathematics in terms of …
- 16.6k
7
votes
Was the early calculus inconsistent?
I would agree with Alexandre Eremenko's answer. The early calculus in fact was not inconsistent, as elaborated below.
Joël's answer is based on a premise that "the question is not precise enough to …
- 16.6k
0
votes
Surreal numbers vs. non-standard analysis
The real question as far as "ordinary mathematics" is concerned is whether there is a set-size surreal extension of the reals useful in doing analysis, and that as a very minimum admits a sine functio …
- 16.6k
2
votes
Was Cauchy prescient?
For convenience of our readers I provide a summary of the article linked in the question:
Cauchy's sum theorem is a prototype of what is today
a basic result on the convergence of a series of fu …
12
votes
How helpful is non-standard analysis?
I just came across a 2013 book by F. Herzberg entitled "Stochastic Calculus with Infinitesimals" where probability and stochastic analysis are done without having to develop the complexities of measur …
- 16.6k
5
votes
Accepted
Are there results unique to non-standard analysis or surreal numbers that have not been reco...
The key paper in this area is Henson and Keisler:
C. W. Henson and H. J. Keisler, On the strength of nonstandard
analysis}, J. Symbolic Logic, 51 (1986), no. 2, 377-386.
The elaborate on the point y …
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