There was a time when there was no such thing as algebra, or calculus, or propositional logic. How were they invented? Look into that question, and you will be investigating how rigor and method was introduced where previously it did not exist.

History of concept of computation seems very analogous to development of concept of justification. I think we're at roughly Leibniz stage of figuring out justification. (I sorta wanna write up a thorough analysis of this somewhere.)