Logical arguments for the death of God.

[Age]

Okay, so absolutely every thing that 'I' say about 'proof', 'proving', or 'verifying proof', which applies to human beings, to you, is 'necessarily wrong'.

That is not what I have said. The following is what I have said:

If what you say about proof only applies to humans and cannot apply to computers, then it is wrong.

Now, it is great that 'you' actually acknowledged that it was 'you' who actually said, 'If what I say about proof only applies to humans and cannot apply to computers', because it is completely blatantly obvious that I have not said absolutely any thing about 'proof only applies to humans and cannot apply to computers'. So, now that 'we' have sorted 'this' out thoroughly, you using the 'if' word, here, is and was just a complete waste of 'time', once more.

I suggest, so as to speed 'the process' up, here, that you only 'look at', and only 'see', only the 'actual words' that I write and use, here, and stop with this persistent assuming'of yours, completely and utterly.

Humans and computers are substitutes for each other as provers and verifiers.

Further to 'this' 'trees', 'mountains', 'screens', and 'words', themselves, can be so-called 'provers' and/or 'verifiers', themselves. All depending on the situation/s, of course.

There is, of course, the sidenote that humans are noticeably better at discovering proof. Computers, on the other hand, are noticeably faster at verifying proof.

Okay, but considering the fact that you believe, and insist, absolutely that you human beings can not produce, and thus can not discover, 'proof', by 'observing' any thing at all, then how do you propose that you human beings 'discover' 'proof', exactly?

I, for One, anyway, look forward to your answer/s, and clarification, here, as always.

This invariant -- humans and computers are interchangeable in the context of proof -- is very important when discussing proof. If a computer cannot possibly verify your proof, then it is not even a proof to begin with.

Is 'this' an irrefutable Fact?

If yes, then how can you 'prove' this, exactly?

This emphasizes an important property of proof. It is both produced and verified by pure reason alone. Computers are blind and deaf. They must still be able to verify your proof. Otherwise, it is not proof.

Once more, 'you' keep making claims, but keep not providing any actual 'proof' for 'your claim'.

The Curry-Howard correspondence is adamant about the following:

A proof is a computation.

So, all it takes is just two people, alone, to be 'adamant' about some thing, then 'you' will take 'that thing' as being 'absolutely irrefutably true'. And, all you needed 'for proof' was just 'those two' being adamant about 'their claim', correct?

[godelian]

Is 'this' an irrefutable Fact?
If yes, then how can you 'prove' this, exactly?

ChatGPT: Since there is no objective method available, neither to accept not to reject a proof that is not formalizable in principle, the requirement of being "formalizable in principle" necessarily follows from the definition of proof as an irrefutable argument.

Yes, your statement is philosophically and logically sound, assuming a specific definition of "proof." Let’s break it down:

"Proof as an irrefutable argument" implies a standard of universality and rigor, where no rational objection is possible.

"Irrefutable" in this sense means that anyone following the rules of logic and the structure of the argument must accept it—this is the hallmark of formal proofs.

If a proof cannot be formalized even in principle, then it cannot meet this standard of irrefutability, because there’s no objective mechanism to verify it beyond subjective or intuitive conviction.

Therefore, by defining proof as irrefutable, the requirement of formalizability in principle is built into the concept itself.

This is closely aligned with the classical (and Hilbertian) view of mathematical proof.

In other words, it is a tautologiy, if proof is meant to be an irrefutable argument.

[godelian]

This emphasizes an important property of proof. It is both produced and verified by pure reason alone. Computers are blind and deaf. They must still be able to verify your proof. Otherwise, it is not proof.

Once more, 'you' keep making claims, but keep not providing any actual 'proof' for 'your claim'.

So, as argued in my previous reply, it is a tautologiy that an irrefutable argument ("proof") must be formalizable in principle.

ChatGPT: Can a fully formalized proof always be verified by a computer?

Yes, a fully formalized proof can always be verified by a computer, provided that:

1. The proof is written in a formal system that the computer understands (such as in the syntax of a proof assistant like Coq, Lean, or Isabelle).

2. The rules of inference and axioms of the system are clearly defined and finite so that the computer can apply them algorithmically.

Verification in this context means checking, step by step, that each inference in the proof follows validly from the previous ones using the formal rules. This is a mechanical, decidable process, assuming the proof is syntactically correct and doesn't require higher-level "intuition."

To clarify: verification is not the same as discovery. Discovering proofs can be undecidable and non-computable in general (due to Gödel’s incompleteness theorems and the undecidability of first-order logic), but checking a given, properly formalized proof is a deterministic task a computer can do.

Would you like an example of how a proof assistant like Lean verifies a theorem?

Proof assistants such as Coq and Isabelle can verify any legitimate proof when expressed in their particular formal syntax. Therefore, these programs are formally constructive witnesses that prove the claim.

Hence, we can conclude that an irrefutable argument ("proof") must be formalizable in principle and can in principle always be verified by a computer. QED.

[Age]

Is 'this' an irrefutable Fact?
If yes, then how can you 'prove' this, exactly?

ChatGPT: Since there is no objective method available, neither to accept not to reject a proof that is not formalizable in principle, the requirement of being "formalizable in principle" necessarily follows from the definition of proof as an irrefutable argument.

Yes, your statement is philosophically and logically sound, assuming a specific definition of "proof." Let’s break it down:

"Proof as an irrefutable argument" implies a standard of universality and rigor, where no rational objection is possible.

"Irrefutable" in this sense means that anyone following the rules of logic and the structure of the argument must accept it—this is the hallmark of formal proofs.

If a proof cannot be formalized even in principle, then it cannot meet this standard of irrefutability, because there’s no objective mechanism to verify it beyond subjective or intuitive conviction.

Therefore, by defining proof as irrefutable, the requirement of formalizability in principle is built into the concept itself.

This is closely aligned with the classical (and Hilbertian) view of mathematical proof.

In other words, it is a tautologiy, if proof is meant to be an irrefutable argument.

If you so believe and say so. But, again, what you are saying as just about absolutely nothing at all to do with what I was saying, pointing out, and meaning, exactly.

[Age]

This emphasizes an important property of proof. It is both produced and verified by pure reason alone. Computers are blind and deaf. They must still be able to verify your proof. Otherwise, it is not proof.

Once more, 'you' keep making claims, but keep not providing any actual 'proof' for 'your claim'.

So, as argued in my previous reply, it is a tautologiy that an irrefutable argument ("proof") must be formalizable in principle.

ChatGPT: Can a fully formalized proof always be verified by a computer?

Yes, a fully formalized proof can always be verified by a computer, provided that:

1. The proof is written in a formal system that the computer understands (such as in the syntax of a proof assistant like Coq, Lean, or Isabelle).

2. The rules of inference and axioms of the system are clearly defined and finite so that the computer can apply them algorithmically.

Verification in this context means checking, step by step, that each inference in the proof follows validly from the previous ones using the formal rules. This is a mechanical, decidable process, assuming the proof is syntactically correct and doesn't require higher-level "intuition."

To clarify: verification is not the same as discovery. Discovering proofs can be undecidable and non-computable in general (due to Gödel’s incompleteness theorems and the undecidability of first-order logic), but checking a given, properly formalized proof is a deterministic task a computer can do.

Would you like an example of how a proof assistant like Lean verifies a theorem?

Proof assistants such as Coq and Isabelle can verify any legitimate proof when expressed in their particular formal syntax. Therefore, these programs are formally constructive witnesses that prove the claim.

Hence, we can conclude that an irrefutable argument ("proof") must be formalizable in principle and can in principle always be verified by a computer. QED.

If this is so, then okay.

But, if this has any thing at all to do with what I was just saying, meaning, and pointing out, then I am not sure what that is, at all.

you claim that you can not produce 'proof' by observing any thing at all. I have just been saying, meaning, and pointing out that, actually, for some people, but obviously not 'you' "godelian" 'proof' can actually be provided, and/or produced, by just the 'observing' of some thing, or another. As others will clearly attest to.

[godelian]

But, if this has any thing at all to do with what I was just saying, and meaning, then I am not sure what that is, at all.

You asked for proof of the following:

If an irrefutable argument ('proof") can in principle not be verified by a computer, then it cannot possibly be irrefutable.

[godelian]

you claim that you can not produce 'proof' by observing any thing at all.

Well, try to encode your observation in the syntax of the Coq proof assistant and let him confirm that it is irrefutable.
If there is simply no way to do that, then you know that it is not irrefutable.

[Age]

But, if this has any thing at all to do with what I was just saying, and meaning, then I am not sure what that is, at all.

You asked for proof of the following:

If an irrefutable argument ('proof") can in principle not be verified by a computer, then it cannot possibly be irrefutable.

When and where, exactly, did I, supposedly, ask for proof of 'that'?

you said and claimed that you can not produce proof by observing any thing at all. I say and claim I can.

Do you comprehend and understand what this means, exactly?

[godelian]

you said and claimed that you can not produce proof by observing any thing at all. I say and claim I can.

You cannot provide an observation to the Coq proof assistant that he will confirm to be an irrefutable argument. It cannot be done. So, no, you cannot.

[Age]

you claim that you can not produce 'proof' by observing any thing at all.

Well, try to encode your observation in the syntax of the Coq proof assistant and let him confirm that it is irrefutable.

But, I am not the one who is 'trying to' get their claim believed to be absolutely true by any one, here.

Also, what does the word 'assistant' mean, or is referring to, to you, exactly?

If there is simply no way to do that, then you know that it is not irrefutable.

But, I do not know this at all.

And, 'you' telling 'me' what I know, or do not know, is never going to work, here.

[Age]

you said and claimed that you can not produce proof by observing any thing at all. I say and claim I can.

You cannot provide an irrefutable argument for the Coq proof assistant to confirm by means of an observation. It cannot be done. So, you cannot.

Are you able to show, and prove, that you can provide an irrefutable argument for some so-labelled 'Coq proof assistant', which will confirm that what you are saying and claiming, here, is an irrefutable argument?

If yes, then will 'you' do this, for 'us'?

If no, then why not? What have you got to hide and/or be afraid of, here, exactly?

[godelian]

Also, what does the word 'assistant' mean, or is referring to, to you, exactly?

The Coq proof "assistant":

https://rocq-prover.org/

A trustworthy, industrial-strength interactive theorem prover and dependently-typed programming language for mechanised reasoning in mathematics, computer science and more.

Latest Rocq Prover release: 9.0.0
Latest Rocq Platform release: 2025.01.0

The Rocq Prover was formerly known as the Coq Proof Assistant (see more on the name evolution).


A short introduction to the Rocq Prover

The Rocq Prover is an interactive theorem prover, or proof assistant. This means that it is designed to develop mathematical proofs, and especially to write formal specifications: programs and proofs that programs comply to their specifications. An interesting additional feature of Rocq is that it can automatically extract executable programs from specifications, as either OCaml or Haskell source code.

In general, what is a "proof assistant":

Gemini AI:

A proof assistant is a software tool that assists in constructing and verifying formal proofs, primarily within mathematics and computer science. It's not about automatically generating proofs, but rather about helping users interactively develop and check the correctness of formal proofs, often using a combination of human guidance and automated reasoning techniques.

You believe that you can provide a physical observation to the Rocq proof assistant for him to confirm that as an irrefutable argument. I keep telling you that you cannot do that. What you want to do, is utterly impossible.

[godelian]

Are you able to show, and prove, that you can provide an irrefutable argument for some so-labelled 'Coq proof assistant', which will confirm that what you are saying and claiming, here, is an irrefutable argument?

If yes, then will 'you' do this, for 'us'?

If no, then why not? What have you got to hide and/or be afraid of, here, exactly?

I can somewhat write candidate theorems in TPTP language, which is suitable for theorem provers such as vampire. This is an example theorem:

$ cat socrates.p

fof(a1, axiom,
man(socrates)).

fof(a2, axiom,
![X] : (man(X) => mortal(X))).

fof(c1, conjecture,
mortal(socrates)).

The following is the output that confirms the status of the example above as a provable theorem:

$ ./vampire socrates.p
% Running in auto input_syntax mode. Trying TPTP
% Refutation found. Thanks to Tanya!
% SZS status Theorem for socrates
% SZS output start Proof for socrates
1. man(socrates) [input]
2. ! [X0] : (man(X0) => mortal(X0)) [input]
3. mortal(socrates) [input]
4. ~mortal(socrates) [negated conjecture 3]
5. ~mortal(socrates) [flattening 4]
6. ! [X0] : (mortal(X0) | ~man(X0)) [ennf transformation 2]
7. man(socrates) [cnf transformation 1]
8. ~man(X0) | mortal(X0) [cnf transformation 6]
9. ~mortal(socrates) [cnf transformation 5]
10. mortal(socrates) [resolution 8,7]
11. $false [subsumption resolution 10,9]
% SZS output end Proof for socrates
% ------------------------------
% Version: Vampire 4.9 (commit 5ad494e78 on 2024-06-14 14:05:27 +0100)
% Linked with Z3 4.12.3.0 79bbbf76d0c123481c8ca05cd3a98939270074d3 z3-4.8.4-7980-g79bbbf76d
% Termination reason: Refutation

% Memory used [KB]: 415
% Time elapsed: 0.010 s
perf_event_open failed (instruction limiting will be disabled): Permission denied
(If you are seeing 'Permission denied' ask your admin to run 'sudo sysctl -w kernel.perf_event_paranoid=-1' for you.)
% ------------------------------
% ------------------------------

If I fully formalize my argument about why it must be possible for a computer to verify irrefutable arguments, because otherwise these arguments are simply not irrefutable, will you even be able to understand the input and output of this Vampire tool? You already do not understand what I say in natural language. Do you really want me to show you some more program input and program output?

Have you ever done any programming in your life, even of the simplest kind?

You see, you have a habit of rejecting arguments because you do not understand them, and not because they would be wrong. So, you are again not going to understand anything whatsoever, and then you will insist that it must all be wrong. Is it really needed to go through that phase again with you?

[Age]

Also, what does the word 'assistant' mean, or is referring to, to you, exactly?

The Coq proof "assistant":

https://rocq-prover.org/

A trustworthy, industrial-strength interactive theorem prover and dependently-typed programming language for mechanised reasoning in mathematics, computer science and more.

Latest Rocq Prover release: 9.0.0
Latest Rocq Platform release: 2025.01.0

The Rocq Prover was formerly known as the Coq Proof Assistant (see more on the name evolution).


A short introduction to the Rocq Prover

The Rocq Prover is an interactive theorem prover, or proof assistant. This means that it is designed to develop mathematical proofs, and especially to write formal specifications: programs and proofs that programs comply to their specifications. An interesting additional feature of Rocq is that it can automatically extract executable programs from specifications, as either OCaml or Haskell source code.

In general, what is a "proof assistant":

Gemini AI:

A proof assistant is a software tool that assists in constructing and verifying formal proofs, primarily within mathematics and computer science. It's not about automatically generating proofs, but rather about helping users interactively develop and check the correctness of formal proofs, often using a combination of human guidance and automated reasoning techniques.

You believe that you can provide a physical observation to the Rocq proof assistant for him to confirm that as an irrefutable argument.

If 'this' is what you, still, are 'currently' believing is true, then 'you' really are an absolute imbecile.

But, if you do not believe that 'this' is true, then why did you say and write 'this', here?

I keep telling you that you cannot do that. What you want to do, is utterly impossible.

1. And I keep informing you that I never ever, ever, said absolutely any such thing.

2. Again, I have never ever, ever, wanted to do such a thing.

And, you consistently telling 'yourself" such things as 'this' is showing and proving just how Truly 'unstable' you really are being, here.

[Age]

Are you able to show, and prove, that you can provide an irrefutable argument for some so-labelled 'Coq proof assistant', which will confirm that what you are saying and claiming, here, is an irrefutable argument?

If yes, then will 'you' do this, for 'us'?

If no, then why not? What have you got to hide and/or be afraid of, here, exactly?

I can somewhat write candidate theorems in TPTP language, which is suitable for theorem provers such as vampire. This is an example theorem:

$ cat socrates.p

fof(a1, axiom,
man(socrates)).

fof(a2, axiom,
![X] : (man(X) => mortal(X))).

fof(c1, conjecture,
mortal(socrates)).

The following is the output that confirms the status of the example above as a provable theorem:

$ ./vampire socrates.p
% Running in auto input_syntax mode. Trying TPTP
% Refutation found. Thanks to Tanya!
% SZS status Theorem for socrates
% SZS output start Proof for socrates
1. man(socrates) [input]
2. ! [X0] : (man(X0) => mortal(X0)) [input]
3. mortal(socrates) [input]
4. ~mortal(socrates) [negated conjecture 3]
5. ~mortal(socrates) [flattening 4]
6. ! [X0] : (mortal(X0) | ~man(X0)) [ennf transformation 2]
7. man(socrates) [cnf transformation 1]
8. ~man(X0) | mortal(X0) [cnf transformation 6]
9. ~mortal(socrates) [cnf transformation 5]
10. mortal(socrates) [resolution 8,7]
11. $false [subsumption resolution 10,9]
% SZS output end Proof for socrates
% ------------------------------
% Version: Vampire 4.9 (commit 5ad494e78 on 2024-06-14 14:05:27 +0100)
% Linked with Z3 4.12.3.0 79bbbf76d0c123481c8ca05cd3a98939270074d3 z3-4.8.4-7980-g79bbbf76d
% Termination reason: Refutation

% Memory used [KB]: 415
% Time elapsed: 0.010 s
perf_event_open failed (instruction limiting will be disabled): Permission denied
(If you are seeing 'Permission denied' ask your admin to run 'sudo sysctl -w kernel.perf_event_paranoid=-1' for you.)
% ------------------------------
% ------------------------------

If I fully formalize my argument about why it must be possible for a computer to verify irrefutable arguments, because otherwise these arguments are simply not irrefutable, will you even be able to understand the input and output of this Vampire tool? You already do not understand what I say in natural language. Do you really want me to show you some more program input and program output?

Have you ever done any programming in your life, even of the simplest kind?

You see, you have a habit of rejecting arguments because you do not understand them, and not because they would be wrong. So, you are again not going to understand anything whatsoever, and then you will insist that it must all be wrong. Is it really needed to go through that phase again with you?

you appear to keep forgetting that it is 'you' who made the claim, you cannot produce 'proof' by observing absolutely any thing at all.

And, you have also completely forgotten that, actually, I showed, and proved, how it is very, very simple and just as easy to produce 'proof' by observing things. Full stop. End of story.