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The method of constructing infinitesimals of the kind used in nonstandard analysis depends on the model and which collection of axioms are used. We consider here systems where infinitesimals can be shown to exist. In 1936 Maltsev proved the compactness theorem. This theorem is fundamental for the existence of infinitesimals as it proves that it ...
The Method takes the form of a letter from Archimedes to Eratosthenes, [1] the chief librarian at the Library of Alexandria, and contains the first attested explicit use of indivisibles (indivisibles are geometric versions of infinitesimals).
During the Hellenistic period, this method was further developed by Archimedes (c. 287 – c. 212 BC), who combined it with a concept of the indivisibles—a precursor to infinitesimals—allowing him to solve several problems now treated by integral calculus.
[1] [6] Despite the benefits described by Sullivan, the vast majority of mathematicians have not adopted infinitesimal methods in their teaching. [7] Recently, Katz & Katz [8] give a positive account of a calculus course based on Keisler's book. O'Donovan also described his experience teaching calculus using infinitesimals.
The key to our method is provided by the detailed analysis of the relation between mathematical languages and mathematical structures which lies at the bottom of contemporary model theory. In 1973, intuitionist Arend Heyting praised nonstandard analysis as "a standard model of important mathematical research".
Keisler's Elementary Calculus: An Infinitesimal Approach defines continuity on page 125 in terms of infinitesimals, to the exclusion of epsilon, delta methods. The derivative is defined on page 45 using infinitesimals rather than an epsilon-delta approach. The integral is defined on page 183 in terms of infinitesimals.
Smooth infinitesimal analysis is a modern reformulation of the calculus in terms of infinitesimals.Based on the ideas of F. W. Lawvere and employing the methods of category theory, it views all functions as being continuous and incapable of being expressed in terms of discrete entities.
The word calculus is a Latin word, meaning originally "small pebble"; as such pebbles were used for calculation, the meaning of the word has evolved and today usually means a method of computation. Meanwhile, calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the study of continuous change.