Oh, so centuries later? Right, just as I said.
> those continental philosophy (as opposed to analytic philosophy).
W. V. O. Quine, one of the most influential analytic philosophers (and mathematician, responsible for an early attempt to formalize mathematical axioms), still felt the need to write against Kant in one of his most influential papers ever (Two Dogmas of Empiricism), well into the 20th century.
> only if we constrain it to formal axiomatc proofs
Umm, no. Inferential systems need not be axiomatic. I was explicitly speaking about inferential systems, not axiomatic.
> there are proofs by example
No, there aren't. There are proofs by contradiction, wherein the truth of a proven statement refutes the candidate-truth of some other hypothetical candidate-truth. But there's no such thing as "proof by example".
> proving by non-axiomatic argumentation and fuzzy evidence (as in a court)
Again, no. Not even judges refer to this as "proof". Evidence is not proof. You're (maybe on purpose?) conflating "deduction" in particular with "reasoning" in general.
> Leibniz's/Russel's/Hilbert's axiomatic mathematical proofs are not the only game in town, nor do they exhaust human reasoning.
Ahhh, there it is. You're mixing up several concepts here. I suspect because you seem to think your pride (i.e. ability to prove things) is on the line. But it isn't. I'm just an autistic spectrum person, trying to make sure people use words in ways that make sense and follow well-established precedent. It appears we have two wildly different projects.