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this post was submitted on 22 Apr 2026
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Exactly.
But at the same time, even if it's true everywhere forever, it might still not be provable, because Gödel.
I think saying that a theorem is true presumes the axioms from which it was proven and so the entire system is "true everywhere forever".
I often find it helpful to think of chess as my axiomatic system. When we say the king is in checkmate, it presumes that we accept all the underlying rules of chess. And these pieces that theoretically form a checkmate will always do so forever... Assuming the usual rules of chess, assuming they're unchanging, etc.
When you put things in terms of chess, these "deep" statements about "math" often become banal. And it works for any game that's a "formal system" (eg. most board games).
No. Gödel's completeness theorem says that if something is true in every model of a (first-order) theory, it must be provable. Gödel's incompleteness theorem says that for every sufficiently powerful theory, there exists statements that are true sometimes, and these can't be provable.
The key word is "everywhere".
Worse: If the chosen axioms are contradictory, then the theorem is effectively worthless.
And it is impossible to know whether axioms are consistent. You can only prove that they are not.
You can go deeper. To prove anything, including the consistency or inconsistency of a theory, you need to work within a different system of axioms, and assume that it is consistent, etc.
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.
Wait do you think Bertrand Russell and Alan Turing and Kurt Gödel weren't making philosophical arguments?
They are clearly mathematical. Starting with definitions and axioms and deriving results from there using mathematical statements.
Sure. But they're also philosophical. The categories aren't mutually exclusive. Basic set theory (which is both mathematics and philosophy).
They all debated the question what being mathematical means there whole lives.
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 already knew that. You're treading an old worn out logical positivist path, that was inspired by Wittgenstein who worked closely with Russell (both mathematicians and philosophers) and he later saw his error, rejected his positivist followers and explained how truth is not a correspondence to facts, rather meaning is derived from use in language. This applies to all languages, formal and informal, including math and logic.
It's the other way around: math is where you just ignore questions about what makes sense, what knowledge is, what truth is, what a proof is, how scientific consensus is reached, what the scientific method should be, and so on. Instead, you just handwave and assume it will all work out somehow.
Philosophy of mathematics is were these questions are treated rigorously.
Of course, serious mathematicians are often philosophers at the same time.
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.
I explicitly refer to your second paragraph.
Yes, you absolutely can argue computer verified proofs. They are very likely to be true (same as truth in biology or sociology: a social construct), but to be certain, you would need to solve the halting problem to proof the program and it's compiler, which is impossible. Proofing incompleteness with computers isn't relevant, because it wasn't in question and it doesn't do away with it's epistemological implications.
It is not necessary to solve the halting problem to show that a particular lean proof is correct.
Lean runs on C++. C++ is a turning complete, compiled language. It and it's compiler are subject to the halting problem.
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's not about those specific proofs. You're claiming, that every possible proof stated in lean will always halt. Lean tries to evade the halting problem best as possible, by requiring functions to terminate before it runs a proof. But it's not able to determine for every lean program it halts or not. That would solve the halting problem. Furthermore, the kernel still relies on CPU, memory and OS behavior to be bug free. Can you be sure enough in practice, yeah probably. But you're claiming absolute metaphysical certainty that abolishes the need for philosophy and sorry, but no software will ever achieve that.
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.
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.
Exactly, I'm glad you understand. There's no epistemological certainty in math, just like in normal language. We have to make do with being pretty certain, as good as it gets. I like lean for it's intended purpose: advancing math. No one involved in lean is seriously claiming it produces some kind of religious absolute certainty. Neither is anyone trying to replace philosophy.
Math can't elevate anything above philosophy, because in a sense, it is part of philosophy, one of the parts using specialized language, specifically the part that is concerned with tautologies.
Have you clicked on the links to the philosophy wiki I provided? Maybe read about what a brilliant mathematician and philosopher has written on the philosophy of mathematics to convince yourself, that philosophy of mathematics is valuable and necessary (wether you agree with his specific point of view or not). You're already engaging in philosophical debate yourself. Your claims about the nature of philosophical arguments and mathematical proofs are themselves philosophical in nature.
Also, though you haven't clearly articulated your philosophical position, it seems to be close to the one of the famous Vienna Circle , which was inspired by Wittgenstein, but later rejected by him. It's generally agreed today, that their project of logical empiricism has failed. You can find the critiques of the various points in the article above.
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.
Yes, they are. Have you seen the controversies around many recent proofs? Proofs are getting so long and topics so specialized, that simply just reading them takes for ever. Some important ones have only been checked by one or two people. Some have been out for years and are still controversial, because no one claims to have some the immense work to actually checked them. That's one of the reasons why proof assistants are used in the first place. They help, but they come with their own problems and challenges.
This is such a very old idea and you're not the first one to have it. Just try it yourself as an exercise. Is like to see how you get an ought from an is with pure math. Every one who tried to build ethics on math only failed. Please, just google it or read some of the links I shared. Philosophers are totally familiar with very advanced math and use it. Again read some articles on like set theory or quantum mechanics on plato.stanford.edu to verify yourself. It's already being used and always has. Even the antique philosophers were mathematicians. They invented logic and geometry. Every philosophy student through antiquity and the middle ages up to the Renaissance was forced to learn them before getting to the more advanced topics.
No matter how smart you are, other smart people probably had very similar ideas before you, tried to formalize them, got challenged, responded, tried again and so on. The history of their work is the history of philosophy. Trying to do better without even reading any of it would fit the definition of being naive.
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.
What do you actually think is philosophy and what do you propose instead? How do you know your "tools" are better? Better by which criteria? Why those and not others? Even just attempting to answer any of these questions is doing philosophy. You can't escape it. Framing philosophical questions in the language of say, set theory, like Russel did, dosn't answer them. It's just using another language. The Vienna Circle thought (inspired by Wittgenstein) that using a formal language would make the answers perfectly clear. And the one who refuted them, proofed them wrong, was no other then the one they admired the most, Wittgenstein himself. No one will take your ideas seriously, if you don't engage with this history first. I'm not saying it's pointless or stupid, it might well be worthwhile. You just have to do it first or end up embarrassingly chasing around the first idea that pops into your head. Like "I feel sure about my answers in a math test and unsure about my essay in philosophy class, that's why math is the best and philosophy is stupid" this is the infantile and emotional level your understanding of both philosophy and math is at currently. Or maybe it isn't, but it sure seems this way, since you haven't clearly articulated your positions, nor made any attempt to formulate an argument for them. Not using normal language and not using mathy language.
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.