Editing Hyperreal number (section)

=== From Leibniz to Robinson ===
When [[Isaac Newton|Newton]] and (more explicitly) [[Gottfried Leibniz|Leibniz]] introduced differentials, they used infinitesimals and these were still regarded as useful by later mathematicians such as [[Leonhard Euler|Euler]] and [[Augustin Louis Cauchy|Cauchy]]. Nonetheless these concepts were from the beginning seen as suspect, notably by [[George Berkeley]].  Berkeley's criticism centered on a perceived shift in hypothesis in the definition of the derivative in terms of infinitesimals (or fluxions), where ''dx'' is assumed to be nonzero at the beginning of the calculation, and to vanish at its conclusion (see [[Ghosts of departed quantities]] for details).  When in the 1800s [[calculus]] was put on a firm footing through the development of the [[(ε, δ)-definition of limit]] by [[Bernard Bolzano|Bolzano]], Cauchy, [[Karl Weierstrass|Weierstrass]], and others, infinitesimals were largely abandoned, though research in [[non-Archimedean field]]s continued (Ehrlich 2006).

However, in the 1960s [[Abraham Robinson]] showed how infinitely large and infinitesimal numbers can be rigorously defined and used to develop the field of [[nonstandard analysis]].<ref name=robinson>{{Citation | last1=Robinson | first1=Abraham | author1-link=Abraham Robinson | title=Non-standard analysis | publisher=[[Princeton University Press]] | isbn=978-0-691-04490-3 | year=1996}}. The classic introduction to nonstandard analysis.</ref> Robinson developed his theory [[Constructive_proof#Non-constructive_proofs|nonconstructively]], using [[model theory]]; however it is possible to proceed using only [[algebra]] and [[topology]], and proving the transfer principle as a consequence of the definitions. In other words hyperreal numbers ''per se'', aside from their use in nonstandard analysis, have no necessary relationship to model theory or first order logic, although they were discovered by the application of model theoretic techniques from logic.  Hyper-real fields were in fact originally introduced by [[Edwin Hewitt|Hewitt]] (1948) by purely algebraic techniques, using an ultrapower construction.