  Infinitesimals are natural products of the human imagination. Their history
goes back to the Greek antiquity. Their role in the calculus and analysis has
seen dramatic ups and downs. They have stimulated strong opinions and even
vitriol. Edwin Hewitt developed hyperreal fields in the 1940s. Abraham
Robinson's infinitesimals date from the 1960s. A noncommutative version of
infinitesimals, due to Alain Connes, has been in use since the 1990s. We review
some of the hyperreal concepts, and compare them with some of the concepts
underlying noncommutative geometry.
