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.)
I'm doing an undergraduate course on the Free Will Theorem, with three lecturers: a mathematician, a physicist, and David Chalmers as the philosopher. The course is a bit pointless, but the company is brilliant. Chalmers is a pretty smart guy. He studied computer science and math as an undergraduate, before "discovering that he could get paid for doing the kind of thinking he was doing for free already". He's friendly; I've been chatting with him after the classes.
So if anyone has any questions for him, if they seem interesting enough I could approach him with them.
Emails to him also work, of course, but discussion in person lets more understanding happen faster. For example, in a short discussion with him I understood his position on consciousness way better than I would have just from reading his papers on the topic.