An Overview of Non-Standard Analysis: Justifying Infinitesimals for Use in Analysis and Calculus Reece Boston, Georgia College Mathematics Major
In the late 17th century, Gottfried Leibniz and Isaac Newton both made the discovery of calculus independently of one another, both taking extremely different approaches. Newton based his calculus on geometric arguments relating to limits, while Leibniz preferred to think in terms of infinitesimal quantities. Despite the efforts of mathematicians for hundreds of years afterwards, Leibniz was never able to justify his manipulation of infinitesimals in his formulas, and so the delta-epsilon formalism of limit became the canonical approach to the subject of calculus and analysis of the real numbers. It was not until 1961 that Abraham Robinson, using the tools from mathematical logic, was able to develop a solid foundation and justification for Leibniz's use of infinitesimal quantities, thus formally founding Non-Standard analysis. We will briefly examine the motivation for using the so called "hyperreals" in calculus and analysis, briefly construct a hyperreal number system, and discproperties of this number system, especially as they apply to calculus and analysis.
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