Adventures in Formalism

Mathematics originates with intuition. But intuition alone can only go so far and formalism develops to handle the more difficult problems. Formalism, however, has its inherent dangers. There are three types of formalism. Type I formalism, exemplified in the work of Euler, is basically heuristic reasoning, the use of familiar reasoning in areas where the reasoning might not or ought not apply. The results include startling successes, and also theorems admitting exceptions. Type II formalism...