rigorous axioms infinitesimal Michel Rolle Bishop Berkeley ghosts of departed quantities The Analyst area infinitesimals
In calculus, foundations refers to the rigorous development of the subject from axioms and definitions.  In early calculus the use of infinitesimal quantities was thought unrigorous, and was fiercely criticized by a number of authors, most notably Michel Rolle and Bishop Berkeley. Berkeley famously described infinitesimals as the ghosts of departed quantities in his book The Analyst in 1734.  Working out a rigorous foundation for calculus occupied mathematicians for much of the century following Newton and Leibniz, and is still to some extent an active area of research today.
