Nonstandard Analysis. A Practical Guide with Applications

The dream of an infinitesimal calculus worthy of the name, that is to say in which dx and dy are infinitesimal numbers, t f(x) dx is a genuine sum of a such numbers, limits are attained (or almost), formulae of type cp(x) = rf(S ,cp(S -T)) dS ,
with T infinitely small furnish the solution of y' =f(x,y) that satisfies (*) , has always been dreamed by mathematicians and such a dream deserves perhaps an epistemological inquiry.
Some other dreams, lesser maybe if compared wi th the achievements of calculus, have haunted the mathematician's imagination and wishful thought: it is the idea of a WOrld where integers can be classified as "large" " "small" or even "indeterminate" without the loss of consistent reasoning, satisfy the induction principle and where the successors of small integers would remain small (**) ; a world where concrete collections, fuzzy perhaps but anyhow not finite, could be gathered in a single fini te set ; a world where continuous functions would be approximated almost perfectly by polynomials of a fixed degree. In such a world, the finite realms could be explored either through the telescope or through the magni- Eying glass in order to gather entirely new pictures. Within such a world, the criteria of rigor set forth by Weierstrass and G8ttingen, interpreted in a two-fold sense would allow for phantasy and metaphor.

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