# Approaches to analysis with infinitesimals following Robinson, Nelson, and others

@article{Fletcher2017ApproachesTA, title={Approaches to analysis with infinitesimals following Robinson, Nelson, and others}, author={Peter Fletcher and Karel Hrbacek and Vladimir Kanovei and Mikhail G. Katz and Claude Lobry and Sam Sanders}, journal={arXiv: Classical Analysis and ODEs}, year={2017} }

This is a survey of several approaches to the framework for working with infinitesimals and infinite numbers, originally developed by Abraham Robinson in the 1960s, and their constructive engagement with the Cantor-Dedekind postulate and the Intended Interpretation hypothesis. We highlight some applications including (1) Loeb's approach to the Lebesgue measure, (2) a radically elementary approach to the vibrating string, (3) true infinitesimal differential geometry. We explore the relation of…

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## References

SHOWING 1-10 OF 201 REFERENCES

Toward a History of Mathematics Focused on Procedures

- Mathematics
- 2016

The proposed formalisations indicate that Robinson’s framework is more helpful in understanding the procedures of the pioneers of mathematical analysis than a Weierstrassian framework, but this does not mean that Fermat, Gregory, Leibniz, Euler, and Cauchy were pre-Robinsonians.

Standard foundations for nonstandard analysis

- MathematicsJournal of Symbolic Logic
- 1992

The objective of this paper is to show that nonstandard concepts have a natural place in the usual (more or less “standard”) set theory, and to argue that this approach improves upon various aspects of hitherto considered systems, while retaining most of their attractive features.

Proofs and Retributions, Or: Why Sarah Can’t Take Limits

- Mathematics
- 2015

The small, the tiny, and the infinitesimal (to quote Paramedic) have been the object of both fascination and vilification for millenia. One of the most vitriolic reviews in mathematics was that…

Who Gave You the Cauchy–Weierstrass Tale? The Dual History of Rigorous Calculus

- Mathematics
- 2012

Cauchy’s contribution to the foundations of analysis is often viewed through the lens of developments that occurred some decades later, namely the formalisation of analysis on the basis of the…

Almost Equal: the Method of Adequality from Diophantus to Fermat and Beyond

- MathematicsPerspectives on Science
- 2013

We analyze some of the main approaches in the literature to the method of ‘adequality’ with which Fermat approached the problems of the calculus, as well as its source in the παρισότης of Diophantus,…

Non-Standard Models in a Broader Perspective

- Philosophy
- 2005

Non-standard models were introduced by Skolem, first for set theory, then for Peano arithmetic. In the former, Skolem found support for an anti-realist view of absolutely uncountable sets. But in the…

Cauchy's Continuum

- MathematicsPerspectives on Science
- 2011

Cauchy's sum theorem of 1821 has been the subject of rival interpretations ever since Robinson proposed a novel reading in the 1960s. Some claim that Cauchy modified the hypothesis of his theorem in…

Nonstandard Analysis

- MathematicsAm. Math. Mon.
- 2005

The construction of nonstandard extensions could provide a rigorous foundation for the use of infinitesimals in basic analysis, and NSA has become an active branch of research in its own right.