Non-standard analysis is a branch of mathematics that formulates analysis using a rigorous notion of an infinitesimal number.
Non-standard analysis was introduced in the early 1960s by the mathematician Abraham Robinson. He wrote:
Robinson argued that this principle of Leibniz's is a precursor of the transfer principle. Robinson continued:
A non-zero element of an ordered field F is infinitesimal if and only if its absolute value is smaller than any element of F of the form 1/n, for n a standard natural number. Ordered fields that have infinitesimal elements are also called non-Archimedean. More generally, non-standard analysis is any form of mathematics that relies on non-standard models and the transfer principle. A field which satisfies the transfer principle for real numbers is a hyperreal field, and non-standard real analysis uses these fields as non-standard models of the real numbers.
Robinson's original approach was based on these non-standard models of the field of real numbers. His classic foundational book on the subject Non-standard Analysis was published in 1966 and is still in print. On page 88, Robinson writes:
Several technical issues must be addressed to develop a calculus of infinitesimals. For example, it is not enough to construct an ordered field with infinitesimals. See the article on hyperreal numbers for a discussion of some of the relevant ideas.
There are at least three reasons to consider non-standard analysis: historical, pedogogical, and technical.
Much of the earliest development of the infinitesimal calculus by Newton and Leibniz was formulated using expressions such as infinitesimal number and vanishing quantity. As noted in the article on hyperreal numbers, these formulations were widely criticized by George Berkeley and others. It was a challenge to develop a consistent theory of analysis using infinitesimals and the first person to do this in a satisfactory way was Abraham Robinson.
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