Section: Application Domains

Formalized mathematics

The use of computing power has dramatically increased for the past decades, in all fields of human activity, including most branches of sciences, causing a general need for reliable computing. It also often lies the base for new interdisciplinary interactions. This is also true for so called pure mathematics. One can remark that Thomas Hales’ proof of Kepler’s conjecture, which is an undoubted result of pure mathematics, relies on computations in order to establish thousands of semi-numerical, semi-symbolic inequalities. This is done using techniques of optimization which are typically coming from applied mathematics and have been developed for very concrete applications, often engineering problems. On the other hand, the complete classification of the finite simple groups, also known as the "enormous theorem", does not rely on any machine computation, but is a huge compound piece of published mathematics. Such level of intricacy also raised a controversy on the level of confidence one should have in the correctness of the whole. We thus see that the computer here contributes to blur the lines between what was traditionally considered “fundamental” or “applied”. In such situations, by providing a common mathematical language, formal proof systems may be the only way to provide a safe join between these various tools, through the formalization of proofs whose correctness is difficult to assess through purely human means.