In the 17th century, Leibniz and Newton invented calculus using infinitesimally small quantities. 200 years later, Bolzano, Cauchy and Weierstraß made calculus rigorous by introducing the modern epsilon-delta-formulation of limits, bereaving mathematics of intuitively appealing objects. Still, another 150 years later, students of physics and engineering are still taught to think in terms of infinitesimals, with or without the warning never to mention them in the presence of a mathematician.
In this talk, I will discuss some ways of rigorously introducing infinitesimals in modern day mathematics, and — if time admits — I will talk about a model theoretic approach by Robinson/Zakon in more detail.