Irrefutable proof that you saw something?

[Age]

And, what will 'you' use to back up and support what 'you' agree with and accept, here?

I don't require to support anything I say,
as I'm not here to convince anyone to agree.
I am expressing my belief,
and it is on others how they react.

Makes it easier for me.

Yes, that is a "witness deposition", if it concerns a fact. It is certainly considered "evidence" but not of the "irrefutable" kind. By the way, evidence generally does not need to be irrefutable. Most evidence actually isn't -- scientific evidence isn't either -- and that is perfectly fine.

Obviously, by definition, 'evidence' is not necessarily 'irrefutable'. whereas, 'proof', by definition, is 'irrefutable'.

[Ben JS]

Yeah, the particular definition I quoted of proof doesn't really capture what is typically meant by proof.

proof:
evidence or argument establishing a fact or the truth of a statement
OR
The validation of a proposition by application of specified rules, as of induction or deduction, to assumptions, axioms, and sequentially derived conclusions

I think those definitions are better.
And with respect to those definitions,
proof cannot be refuted - though it can be ignored / denied.

[godelian]

Yeah, the particular definition I quoted of proof doesn't really capture what is typically meant by proof.

proof:
evidence or argument establishing a fact or the truth of a statement
OR
The validation of a proposition by application of specified rules, as of induction or deduction, to assumptions, axioms, and sequentially derived conclusions

I think those definitions are better.
And with respect to those definitions,
proof cannot be refuted - though it can be ignored / denied.

Yes, I've given up on using the term "proof" for "irrefutable evidence". Too many people insist on using it for just "evidence". Even ChatGPT does that! So, I've switched to using the adjective "irrefutable" instead.

[godelian]

Obviously, by definition, 'evidence' is not necessarily 'irrefutable'. whereas, 'proof', by definition, is 'irrefutable'.

Technically, yes. Colloquially, unfortunately not. It is the adjective "irrefutable" that does the trick:

ChatGPT: Is it possible to produce irrefutable evidence that you saw something?

In most cases, it's not possible to produce truly irrefutable evidence that you saw something—because claims about perception are inherently subjective and depend on trust in both the observer and the medium of proof.

However, you can get very strong evidence, such as:

Video from a body cam or head-mounted camera, ideally with timestamp and GPS data.

Corroborating witnesses who independently report the same thing.

Physical evidence linked to what you saw (e.g., if you saw a rare animal and collected fur, droppings, or tracks).

Biometric data, like eye-tracking or neural activity recordings (though this is still developing and far from foolproof).

Even then, people can always question the authenticity of recordings, the accuracy of memory, or potential hallucination or fabrication.

So in short: no, not irrefutable—but yes, you can come close depending on the context and available technology.

It often does not matter that the evidence is refutable. It is often still good enough.

[Age]

Obviously, by definition, 'evidence' is not necessarily 'irrefutable'. whereas, 'proof', by definition, is 'irrefutable'.

Technically, yes.

Finally, 'we' got HERE.

Colloquially, unfortunately not.

This, here, is a 'philosophy forum' and not a 'night out', with friends.

'We' are not with family nor work colleagues just discussing what are called 'everyday' or mundane things, here.

Again, if absolutely any one wants to express views and/or make claims, here, in a 'philosophy forum', then I will once more suggest that you have the actual proof that will back up and support, irrefutably, your views, beliefs and claims, here, because there is more chances than not that you will get critiqued, questioned, and/or challenged over what you say and write, here.

After all this, here, is a 'philosophy forum'.

It is the adjective "irrefutable" that does the trick:

ChatGPT: Is it possible to produce irrefutable evidence that you saw something?

In most cases, it's not possible to produce truly irrefutable evidence that you saw something—because claims about perception are inherently subjective and depend on trust in both the observer and the medium of proof.

However, you can get very strong evidence, such as:

Video from a body cam or head-mounted camera, ideally with timestamp and GPS data.

Corroborating witnesses who independently report the same thing.

Physical evidence linked to what you saw (e.g., if you saw a rare animal and collected fur, droppings, or tracks).

Biometric data, like eye-tracking or neural activity recordings (though this is still developing and far from foolproof).

Even then, people can always question the authenticity of recordings, the accuracy of memory, or potential hallucination or fabrication.

So in short: no, not irrefutable—but yes, you can come close depending on the context and available technology.

It often does not matter that the evidence is refutable. It is often still good enough.

'Good enough' for 'what', exactly?

Also, what you are talking about above, here, is nothing whatsoever ever at all that I have been talking about.

[Gary Childress]

Look around you.

You may, for example, see a chair or a table or something else. Now take a piece of paper and write down an irrefutable argument that you did see what you saw. Will the verifier of your argument consider it to be irrefutable?

No, the verifier won't.

What you have seen is true (to you) but unprovable (to others).

Is what we see provable to ourselves? Or how is it that someone would refute what we say we saw? I mean, the options seem to be that someone could think that we are lying or that we are hallucinating. But if it is not the case that we are lying or hallucinating, then to say I saw something seems like it would serve as "irrefutable proof".

In other words, the physical universe is entirely Godelian, i.e. true but not provable.

The abstract Platonic universe of mathematical objects, on the other hand, is overwhelmingly Godelian but not entirely. Some of it, is both true and provable.

True but not provable, is the norm, and not the exception.

If it's unprovable, then how do we know it's "true"? If we don't know if it's "true", then it may not be the case that the universe is "true but not provable". It would only be unprovable as far as we know.

[godelian]

If it's unprovable, then how do we know it's "true"?

In the physical universe, we know that it is true because we can observe it. We cannot prove that we can see it, but it is still true. In the abstract, Platonic universe, we can only use pure reason, which is deaf and blind. Therefore, the standard case is that we cannot know that it is true unless we can prove it.

But then again, even in the abstract, Platonic universe, there are edge cases where we can still see that a statement is true but unprovable. Only three such edge cases are currently known:

https://en.wikipedia.org/wiki/Goodstein%27s_theorem

In mathematical logic, Goodstein's theorem is a statement about the natural numbers, proved by Reuben Goodstein in 1944, which states that every Goodstein sequence (as defined below) eventually terminates at 0. Laurence Kirby and Jeff Paris[1] showed that it is unprovable in Peano arithmetic (but it can be proven in stronger systems, such as second-order arithmetic or Zermelo-Fraenkel set theory). This was the third example of a true statement about natural numbers that is unprovable in Peano arithmetic, after the examples provided by Gödel's incompleteness theorem and Gerhard Gentzen's 1943 direct proof of the unprovability of ε0-induction in Peano arithmetic. The Paris–Harrington theorem gave another example.

Set theory (ZFC) can prove arithmetical sentences that are true in the standard model of Peano arithmetic (PA), which however, may not be provable from PA. This is the explanation:

ChatGPT: If ZFC proves an arithmetical sentence, is it true in the standard model of PA?

Yes, if ZFC proves an arithmetical sentence, then it is true in the standard model of Peano Arithmetic (PA).

Explanation:

An arithmetical sentence is a sentence involving only quantification over natural numbers and the language of arithmetic (addition, multiplication, equality, etc.)—i.e., it belongs to the first-order language of arithmetic.

The key points are:

ZFC is a conservative extension of PA for arithmetical sentences.
This means: If ZFC proves an arithmetical sentence, then that sentence is already true in the standard model of PA (ℕ).
Even though ZFC is much stronger than PA and can talk about sets and infinite cardinals, it still respects the truths of arithmetic.

The standard model of arithmetic is absolute for arithmetical truths.
Truths about the natural numbers (i.e., the standard model ℕ) are not affected by stronger set-theoretic assumptions, because arithmetic is absolute between transitive models of ZFC and the real ℕ.

Soundness of ZFC.
If ZFC is consistent, then everything it proves is true in all of its models. And since ZFC proves true arithmetical sentences only, those must be true in the standard model ℕ.
If ZFC were inconsistent, of course, it could prove false statements. But under the usual assumption that ZFC is consistent, this isn't a problem.

Conclusion:
Yes, assuming ZFC is consistent, any arithmetical sentence it proves is true in the standard model of arithmetic (ℕ).

Would you like a more technical explanation involving reflection theorems or model theory?

[Gary Childress]

If it's unprovable, then how do we know it's "true"?

In the physical universe, we know that it is true because we can observe it. We cannot prove that we can see it, but it is still true. In the abstract, Platonic universe, we can only use pure reason, which is deaf and blind. Therefore, the standard case is that we cannot know that it is true unless we can prove it.

But then again, even in the abstract, Platonic universe, there are edge cases where we can still see that a statement is true but unprovable. Only three such edge cases are currently known:

https://en.wikipedia.org/wiki/Goodstein%27s_theorem

In mathematical logic, Goodstein's theorem is a statement about the natural numbers, proved by Reuben Goodstein in 1944, which states that every Goodstein sequence (as defined below) eventually terminates at 0. Laurence Kirby and Jeff Paris[1] showed that it is unprovable in Peano arithmetic (but it can be proven in stronger systems, such as second-order arithmetic or Zermelo-Fraenkel set theory). This was the third example of a true statement about natural numbers that is unprovable in Peano arithmetic, after the examples provided by Gödel's incompleteness theorem and Gerhard Gentzen's 1943 direct proof of the unprovability of ε0-induction in Peano arithmetic. The Paris–Harrington theorem gave another example.

Set theory (ZFC) can prove arithmetical sentences that are true in the standard model of Peano arithmetic (PA), which however, may not be provable from PA. This is the explanation:

ChatGPT: If ZFC proves an arithmetical sentence, is it true in the standard model of PA?

Yes, if ZFC proves an arithmetical sentence, then it is true in the standard model of Peano Arithmetic (PA).

Explanation:

An arithmetical sentence is a sentence involving only quantification over natural numbers and the language of arithmetic (addition, multiplication, equality, etc.)—i.e., it belongs to the first-order language of arithmetic.

The key points are:

ZFC is a conservative extension of PA for arithmetical sentences.
This means: If ZFC proves an arithmetical sentence, then that sentence is already true in the standard model of PA (ℕ).
Even though ZFC is much stronger than PA and can talk about sets and infinite cardinals, it still respects the truths of arithmetic.

The standard model of arithmetic is absolute for arithmetical truths.
Truths about the natural numbers (i.e., the standard model ℕ) are not affected by stronger set-theoretic assumptions, because arithmetic is absolute between transitive models of ZFC and the real ℕ.

Soundness of ZFC.
If ZFC is consistent, then everything it proves is true in all of its models. And since ZFC proves true arithmetical sentences only, those must be true in the standard model ℕ.
If ZFC were inconsistent, of course, it could prove false statements. But under the usual assumption that ZFC is consistent, this isn't a problem.

Conclusion:
Yes, assuming ZFC is consistent, any arithmetical sentence it proves is true in the standard model of arithmetic (ℕ).

Would you like a more technical explanation involving reflection theorems or model theory?

I'm not seeing where you are proving that the universe is "true but unprovable". Do you mean it's unprovable in the sense that if I were to say I saw the sun rise earlier than normal today, I cannot prove it to anyone else? I mean, if the sun rises at 3 am tomorrow, then it could have been observed with scientific instruments and "proven" to have risen at 3am (perhaps it could even be calculated based on geometry if it isn't the product of some kind of "miracle"). Does Goedel's theorem just mean that if someone contests something that is based on eyewitness testimony, then there is no way to "irrefutably" prove it to that individual person based on eyewitness testimony?

[godelian]

I'm not seeing where you are proving that the universe is "true but unprovable". Do you mean it's unprovable in the sense that if I were to say I saw the sun rise earlier than normal today, I cannot prove it to anyone else?

Yes. The scenario has 3 elements: the prover, the irrefutable argument, and the verifier. What information are you going to provide to the verifier that irrefutably proves to him that "you saw the sun rise earlier than normal today"?

It could have been observed with scientific instruments and "proven" to have risen at 3am

That would be solid evidence but not irrefutable.

Does Goedel's theorem just mean that if someone contests something that is based on eyewitness testimony, then there is no way to "irrefutably" prove it to that individual person based on eyewitness testimony?

Gödel's theorem does not say anything about the physical universe, if only, because mathematics never says anything about the physical universe.

[Gary Childress]

I'm not seeing where you are proving that the universe is "true but unprovable". Do you mean it's unprovable in the sense that if I were to say I saw the sun rise earlier than normal today, I cannot prove it to anyone else?

Yes. The scenario has 3 elements: the prover, the irrefutable argument, and the verifier. What information are you going to provide to the verifier that irrefutably proves to him that "you saw the sun rise earlier than normal today"?

It could have been observed with scientific instruments and "proven" to have risen at 3am

That would be solid evidence but not irrefutable.

Does Goedel's theorem just mean that if someone contests something that is based on eyewitness testimony, then there is no way to "irrefutably" prove it to that individual person based on eyewitness testimony?

Gödel's theorem does not say anything about the physical universe, if only, because mathematics never says anything about the physical universe.

If I saw the sun rise earlier today than normal but it actually didn't, then it would seem that I am either lying or in error in some way. Either the sun did rise earlier or it did not. If the other person were standing next to me when it happened or witnessed the same thing, then it is either proven ot the two of us or the other option is that it's a lie or an error on the part of both of us. If we both see something but don't believe it's "true" then it is not necessarily the case that what we saw is both true and unprovable. It's only the case that X is unprovable as far as we both know, but we cannot say anything about the truth value, because if it is possible that I am hallucinating or maybe my clock was wrong, then I can't know myself either if the sun really did rise earlier than normal. And if I am simply lying, then it is not true period that I saw the sun rise early.

[Gary Childress]

Something similar might apply to mathematics. If I knew mathematics and was trying to prove something to someone who did not know mathematics (was not a witness to the same rules I learned) I wouldn't be able to prove it to them unless I made them a witness also by teaching them mathematics. It must be possible for the verifier to verify for himself. If he can't, then it's not the case that it is "unprovable", only that the verifier is ignorant of something. It would be provable if the verifier knew mathematics or if the verifier were also a witness to the sun rising earlier.

[Age]

Something similar might apply to mathematics. If I knew mathematics and was trying to prove something to someone who did not know mathematics (was not a witness to the same rules I learned) I wouldn't be able to prove it to them unless I made them a witness also by teaching them mathematics. It must be possible for the verifier to verify for himself. If he can't, then it's not the case that it is "unprovable", only that the verifier is ignorant of something. It would be provable if the verifier knew mathematics or if the verifier were also a witness to the sun rising earlier.

your posts in this thread "gary childress" are straight to the point, and counter and/or refute very succinctly, well to me anyway.

Now might be a good time to ask the question,
'How is 'truth', itself, actually obtained?'

And then when 'that answer' becomes also known and fully understood, then 'we' onto 'the question',
''What is 'proof', itself, and, how is 'proof' actually obtained?'

[godelian]

Something similar might apply to mathematics. If I knew mathematics and was trying to prove something to someone who did not know mathematics (was not a witness to the same rules I learned) I wouldn't be able to prove it to them unless I made them a witness also by teaching them mathematics. It must be possible for the verifier to verify for himself. If he can't, then it's not the case that it is "unprovable", only that the verifier is ignorant of something. It would be provable if the verifier knew mathematics or if the verifier were also a witness to the sun rising earlier.

Yes, both proving and verifying proof are algorithms that the person (or the computer) needs to master. It would indeed be pointless to give proof to someone who cannot verify it.

[Age]

Something similar might apply to mathematics. If I knew mathematics and was trying to prove something to someone who did not know mathematics (was not a witness to the same rules I learned) I wouldn't be able to prove it to them unless I made them a witness also by teaching them mathematics. It must be possible for the verifier to verify for himself. If he can't, then it's not the case that it is "unprovable", only that the verifier is ignorant of something. It would be provable if the verifier knew mathematics or if the verifier were also a witness to the sun rising earlier.

Yes, both proving and verifying proof are algorithms that the person (or the computer) needs to master. It would indeed be pointless to give proof to someone who cannot verify it.

For example, it would, indeed, be pointless to even 'try to' give 'the proof' to some one/thing that 'you' saw some thing. For there is only one who can verify this, for sure, and absolutely.

[alan1000]

And all of this is thought to be a debate in mathematics, is it? I guess the moderators are asleep, which would be par for the course.