>and Kant would put the nail in the coffin of the rationalist-empiricist debate in the next century (with his earth-shattering Critique of Pure Reason)
Kant was mostly convincing for himself (to whom he was a great fan of) and Kantians. His arguments were hardly definitive.
>Some things (even true ones!) are simply unprovable
Within the context of a system with certain algebraic properties.
> Kant was mostly convincing for himself (to whom he was a great fan of) and Kantians
This is shown to be false by the historical record. Kant was, unquestionably, the most influential thinker of his day.
Centuries later, Einstein himself felt that it was necessary to write about how Kant was wrong (given modern physics and general relativity), which shows how convincing and influential Kant's arguments were; there's no point to debunking things no one believes.
You're probably looking at Kant with contemporary eyes, thereby deciding that he's obviously wrong and completely unconvincing. But no one looked at Kant with contemporary eyes until contemporary times, and there's a very long road of extremely important and influential scientists/mathematicians/philosophers replying to Kant; which is how our contemporary perspective came to exist.
>> Some things (even true ones!) are simply unprovable
> Within the context of a system with certain algebraic properties.
Right, that's what "provable" means. There's no such thing as a proof, outside of formal inferential systems.
>You're probably looking at Kant with contemporary eyes, thereby deciding that he's obviously wrong and completely unconvincing
Rather with late 19-th and 20th century eyes, those continental philosophy (as opposed to analytic philosophy).
>There's no such thing as a proof, outside of formal inferential systems
Well, only if we constrain it to formal axiomatc proofs. But there are exhaustive proofs (which don't need axiomatic steps), as there are proofs by example, and even proving by non-axiomatic argumentation and fuzzy evidence (as in a court). Leibniz's/Russel's/Hilbert's axiomatic mathematical proofs are not the only game in town, nor do they exhaust human reasoning.
> 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.
>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).
> Within the context of a system with certain algebraic properties.
This downplays the importance of the set of systems for which his proof holds and makes it sound like it applies to some obscure branch of mathematics.
It applies to a huge set of important systems, not least of which is any system that is sufficiently expressive as to uniquely identify the natural numbers.
Kant was mostly convincing for himself (to whom he was a great fan of) and Kantians. His arguments were hardly definitive.
>Some things (even true ones!) are simply unprovable
Within the context of a system with certain algebraic properties.