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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.Yes, because Kant was pre-analytical, so ultimately not compatible with the analytical program as it developed in the 20th century. But the contintental side started the putdown quite earlier (of course from a different perspective).
>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".
Any statement to the effect of "A thing with X, Y, Z, ... properties exists" can be proven by an example of such a thing. "There are numbers larger than 9 divisible by 2" Yes, here is 10 or 56 or .... Any Y that fulfills both those properties proves that the statement is true. Those types of proofs are called "existential proofs".
>Not even judges refer to this as "proof"
Ever heard the phrase "prove beyond any reasonable doubt"? "Burden of proof"? All terms used in law across many cultures, including the US.
>Inferential systems need not be axiomatic. I was explicitly speaking about inferential systems, not axiomatic.
You might have, but Godel's proof (which we were discussing) is formulated on axiomatic systems though.
>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.
Well, and I'm a fellow dweller trying to make sure we don't miss the forest for the trees (as we oft to do), and don't constrain human expression, including the use of words and the ability to prove something, into artificial constraints or some strict adherence to a narrow concept of officialdom ("well-established precedent").
Keep in mind that this part of the subthread started with my answer to the praise of Kant, whose philosophy was quite far from using a "formal inferential system". My original point was about how Kant was only a definitive influece to a certain lineage of thinkers.
Then came a answer to clarify what Godel proved. My point there was that his proof applies to axiomatic systems with certain algebraic properties, not to every system of reasoning. It's a common trope in pseudo-science to take Godel's incompleteness theorems to apply way beyond their scope (even in social matters).
It's under that light that I described alternative ways people prove things, to showcase that this limitation doesn't apply anywhere something is proven (to a societal satisfactory way, not necessarily formal proofs).