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).
even if it’s true everywhere forever, it might still not be provable, because Gödel.
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.
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.
I see philosophy as a place to make nonrigorous arguments.
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.
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.
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.
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.
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.
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.
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.
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.