A course given, summer 2009, at the Nesin Matematik Köyü. The abstract that I submitted was as follows:
Invented in the 17th century if not earlier, calculus can be understood in terms of “infinitesimals”: non-zero numbers whose absolute values are less than every fraction 1/n. In this understanding, the region bounded by a curve is the sum of rectangles of infinitesimal width; the slope of a curve at a point is a ratio of infinitesimals. But there are no infinitesimals on the so-called “real number line”. In the usual “rigorous” treatment of calculus, invented in the 19th century, infinitesimals do not appear: they are replaced with the notion of a “limit”. However, 20th-century logic shows that infinitesimals can be made just as “real” as the real numbers, so that the original intuitive approach to calculus is entirely justified.