And again, the history of philosophy is replacing philosophical arguments with better tools. Your link just shows sloppy thinking from both Hume and his critics.
If a mathematical proof hasn’t been verified, it isn’t accepted. For a proof that uses lots of new nontrivial machinery, the mathematician is expected to give talks to motivate that machinery and answer questions from other mathematicians. Or they can just build their proof in Lean from already well understood axioms.
It’s generally agreed
That’s my point. Mathematical proofs aren’t generally agreed. They are agreed by everyone to logically follow from the definitions and axioms started with. Every single statement in a mathematical proof evaluates to true or false, and if you don’t believe a mathematical proof, you can directly point to a statement that is false. Philosophical arguments are “generally agreed” upon until the tools to take them out of philosophy are developed, and then the philosophical arguments are discarded entirely.
Your same argument that mathematics can be discussed under philosophy can be used to argue that mathematics can be discussed under the framework of wild untethered speculation. Neither one is a convincing argument that philosophy or wild untethered speculation is useful.
This is why ethics has failed. It has been built on the unstable foundation of philosophy instead of on the solid foundation of mathematics.
It’s not about those specific proofs.
It certainly is about those specific proofs and anything that has been rigorously proven in Lean. We’re discussing techniques that show something is correct forever, and those proofs show that something is correct forever. Philosophical arguments don’t even show that something is correct today. This is why the examples I gave earlier are now not explained by philosophy but by other systems. Once the tooling exists to lift a discussion out of philosophy, that is the end of philosophical debate for that topic.
Furthermore, the kernel still relies on CPU, memory and OS behavior to be bug free.
Only to a point, just like human language proofs require the reviewers brains to be bug free to a point. The repeated verification makes proofs as correct as anything can get.
The fact that C++ is Turing complete does not prevent it from computing that 1+1=2. Similarly, the fact that C++ is Turing complete does not prevent programs created from it from verifying the proofs that they have verified. The proof of the halting problem (and incompleteness proofs based on the halting problem) itself halts. https://leanprover-community.github.io/mathlib_docs/computability/halting.html
It is not necessary to solve the halting problem to show that a particular lean proof is correct.
You’re just covering my third paragraph. Yes, everybody is a philosopher because we don’t have the tools to do away with philosophical arguments entirely yet.
Once a mathematical proof has been verified by computer, there is no arguing that it is wrong. The definitions and axioms directly lead to the proved result. There is no such thing as verifying a philosophical argument, so we develop tools to lift philosophical arguments into more rigorous systems. As I’ve shown earlier, and as another commenter added to with incompleteness, this is a common pattern in the history of philosophy.
And we determined that the resulting incompleteness proofs are valid mathematical proofs whose logical correctness has been verified by computer. https://formalizedformallogic.github.io/Catalogue/Arithmetic/G___del___s-First-Incompleteness-Theorem/#goedel-1
They are clearly mathematical. Starting with definitions and axioms and deriving results from there using mathematical statements.
But that’s math. And its proof is math. And that proof is true everywhere forever.
I see philosophy as a place to make nonrigorous arguments. Eventually, other fields advance enough to do away with many philosophical arguments, like whether matter is infinitely divisible or whether the physical brain or some metaphysical spirit determines our actions.
Since this is a question that math hasn’t advanced enough to answer, we can have a philosophical argument about whether other fields will eventually advance enough to get rid of all philosophical arguments.
Better by the only criteria that matters. Once something is proved, everybody will agree to it given enough time to examine and question the proof. Once someone makes a mathematical proof, the philosophical arguments are thrown on the trash heap. As you mentioned, Wittgenstein threw his earlier philosophical arguments on the trash heap. Given a few more years, he would have thrown his latest philosophical arguments on the trash heap as well.