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.
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 why ethics has failed. It has been built on the unstable foundation of philosophy instead of on the solid foundation of mathematics.
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.
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.