Differential and integral calculus – nowadays referred to together as simply “calculus” – were developed in the late 1600s and early 1700s to allow infinitely small numbers and formulas with infinitely many terms. These techniques turned out to be immensely powerful, and it is impossible to imagine modern physics, engineering or mathematics without them. However, for almost two hundred years the theory was plagued by inconsistencies and a complete lack of logical foundation – it gave correct answers when it had no right to do so. In the mid 1800s, a logical framework was constructed which put calculus on a solid footing, and which allowed it to be greatly extended and melded with linear algebra and topology in the twentieth century, becoming even more powerful. (This process continues even today: physicists are finding startling results with “functional integrals”, apparently by magic, but no one yet knows how to logically justify their methods.)
This class will introduce real analysis in its modern form, with motivation from history.