In mathematics, nonstandard calculus is the modern application of infinitesimals, in the sense of nonstandard analysis, to infinitesimal calculus. It provides a rigorous justification for some arguments in calculus that were previously considered merely heuristic.
Non-rigorous calculations with infinitesimals were widely used before Karl Weierstrass sought to replace them with the (ε, δ)-definition of limit starting in the 1870s. (See history of calculus.) For almost one hundred years thereafter, mathematicians such as Richard Courant viewed infinitesimals as being naive and vague or meaningless.[1]
Contrary to such views, Abraham Robinson showed in 1960 that infinitesimals are precise, clear, and meaningful, building upon work by Edwin Hewitt and Jerzy Łoś. According to Howard Keisler, "Robinson solved a three hundred year old problem by giving a precise treatment of infinitesimals. Robinson's achievement will probably rank as one of the major mathematical advances of the twentieth century."[2]
^Courant described infinitesimals on page 81 of Differential and Integral Calculus, Vol I, as "devoid of any clear meaning" and "naive befogging". Similarly on page 101, Courant described them as "incompatible with the clarity of ideas demanded in mathematics", "entirely meaningless", "fog which hung round the foundations", and a "hazy idea".
^Elementary Calculus: An Infinitesimal Approach, p. iv.
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