User blog comment:KurohaKafka/The Strength of Type Theory/@comment-32876686-20171020190557/@comment-1605058-20171020213538

It is a particular kind of a formal system in which every expression (called a term) is assigned a type (one term can have many types). The terms in such a system are subject to conversion/reduction rules, which tell us that terms of certain type are "the same as" (equivalent to, or however you want to interpret it) other terms.

Type theories can be used as an alternative foundation of mathematics, distinct from more common systems built using predicate logic.