A Very Odd Question

[Canute]

My aplogies for the previous mess-up. Here goes with the proper question.

I have been struggling with this question for a while, but think I have an answer. Before posting my answer for comment I would like to know what you think.

This isn't a trap - it's just damn complicated (to me anyway). The question relates to eastern philosophy, which commonly asserts that a proof of anything is impossible (since truth/falsity is a dualistic notion), and Goedel's Incompleteness theorems, which suggest something rather similar (although people disagree as to the extent of the similarity).

 

[gendanken]

Hmph. I read this three times and still don't get your question. Are you marveling at the spooky congrunce between East and West? Something like one half foreshadowing the theologies of the other in complete ignorance of the other, i.e. pyramid syndrome?

Or do you just aim to sit and muse on how fruitless proving things are, have, and ever will be?

 

[wesmorris]

Yes, good question... definately not odd in my opinion, though I'm not sure my answer isn't silly.

I said false, but now I'm thinking I'm wrong. I think it's set up to fail. You should rephrase it IMO. "things can only be proven subjectively" for instance.. or "there exists some stuff that can be proven, but it seems that most things that humans think about cannot be" or something.

 

[Xenu]

To say "True" is to say that your statement is provable, and then therefore it is true to prove something. Therefore, you run into a paradox.

To say "False" is to say that the statement isn't true. However, if my logic serves me correctly (it's been a while), saying "False" doesn't mean that it is necessarily possible to prove something.

My answer is "Mu", which is an Eastern answer that in this case says that the nature of the question is wrong.

 

[Canute]

Originally posted by Xenu
To say "True" is to say that your statement is provable, and then therefore it is true to prove something. Therefore, you run into a paradox.

To say "False" is to say that the statement isn't true. However, if my logic serves me correctly (it's been a while), saying "False" doesn't mean that it is necessarily possible to prove something.

My answer is "Mu", which is an Eastern answer that in this case says that the nature of the question is wrong.

I take that as 'undecidable' so I've ticked that box on your behalf. I won't comment for now.

 

[lifegazer]

Originally posted by Canute
The question relates to eastern philosophy, which commonly asserts that a proof of anything is impossible (since truth/falsity is a dualistic notion), and Goedel's Incompleteness theorems, which suggest something rather similar (although people disagree as to the extent of the similarity).

A proof of what? That a statement is true?
I find such arguments - devised via thought - to be absolutely nonsensical.
I have alot of respect for Eastern religions, but reason of this kind makes a mockery of reason. It's just like saying "There are no absolute truths" - and saying that this is absolutely so. Therefore, it's a complete mis-use of logic, and has no place in a philosophy forum. Sorry, but that's how I feel.

 

[gendanken]

Says lifegazer:

..... but reason of this kind makes a mockery of reason. It's just like saying "There are no absolute truths" - and saying that this is absolutely so.

Occidental thinking has been just as fired up insisisting on absolutes and has made mess out of things too.

The easterns are o.k. with I-don't-knows and not-for-sures while the Western mind simply will not rest until it feels it knows knows. And somehow the east loses out in your book?

SOoooo...what, pray tell, are you talking about?

 

[lifegazer]

Occidental thinking has been just as fired up insisisting on absolutes and has made mess out of things too.

Having the ability to see impossibilities - so that we can actually see (envisage) the possible nature of existence - opens us up to making mistakes of reason, since there seems to be a couple of possible causes for existence, allowing for a diversity of opinion.
And reason - to this date - has not shown why transforming matter can be its own cause. But these past mistakes/ignorances do not prevent us from ever coming-across a perfectly-reasoned argument showing why transforming-matter
cannot actually be its own absolute-cause.
You cannot say that all philosophy of the absolutes is doomed to being absolutely-awful, absolutely. The past does not dictate the future, essentially.

The easterns are o.k. with I-don't-knows and not-for-sures

The eastern religions are great, imo. But like all religions, they fall- short of making perfect sense. Here though, you give the impression that the eastern religions are okay simply because they admit that they do not know [something] about existence.
My point is that this admitance should mirror the demise of those particular philosophies. For how can we trust a religion that does not know certain things? How can that religion know that anything which it says is absolutely true, unless those truths are related to knowledge of the Absolute?

while the Western mind simply will not rest until it feels it knows knows. And somehow the east loses out in your book?

The East is closest to the truth; but the West has the best attitude towards finding the truth. For the minds in Western culture are more malleable than other minds in the East, generally. In the West, you have a greater chance of having your own mind.
East & West are not in conflict. One has the freedom/power to effect change. And the other has wisdom.
Essentially, the West needs the East's pure use of reason as much as the East needs the West's pure open-mindedness. When that happens, East shall surely meet West.

SOoooo...what, pray tell, are you talking about?

East meets West where reason meets the freedom to reason.
Eastern religions speak alot of wisdom/reason - yet are constricted by context in what they shall reason. Western ~religions~ - namely, materialism - suffer from the same fate. I.e., materialists are no-more convincing than Idealists, whether they have a phd in physics, or not. The belief in an external-reality (external to inner-awareness), is exactly that: a belief. And having a phd in phsics doesn't qualify you as a philosopher of reality.

 

[sparkle]

Falsefalsefalse. DE-FI-NI-TELY False!
Arrived there by two ways:
1. Ruling out. Ruled out “true” for obvious problems with logic and “undecidable” for personal reasons (don’t like undecidable things).
2. Finding examples. I tried to construct situations where you would be able/unable/maybeboth to prove anything impossible. Had a little problem here: definitions. All involved must agree on the very part of (objective?) existence they are going to disprove. Once I had found three examples where EVERY human being agrees to the same terms I had found three examples that could not be disproved. Voila! False.

Your question reminds me: there is an old Chinese story where an emperor took a small struggling bird in his hand and asked a philosopher to tell him, without lying, whether this bird is alive or not.

 

[Canute]

I shouldn't have mentioned Eastern philosophy. There is nothing mystical about the question. It's just an interesting question because it seems to be logically very complicated. I've been thinking about it for a while and I wondered what somebody else would make of it.

It is just as much related to mathematics as it is to metaphysics. Goedel’s incompleteness theorems are often interpreted as proving that we can never completely prove anything. (Eg Hawking seems to take this view).

Here is my attempted interpretation. If it does your head in I apologise, it's certainly done mine in. Believe it or not I've tried to write clearly.

The assertion was:

It is impossible to prove that ‘it is impossible to prove anything’.

For logical reasons it is impossible to prove that this assertion is false, for this would entail that it is possible to prove that all proof is impossible, which is completely self-contradictory. Any such proof would defeat itself.

Because of this the assertion seems bound to be be true, and thus that it is possible, in theory at least, to completely prove some things.

It follows that although Goedel’s theorems do place a restriction on the types of proof that we can use when trying to prove something, and the types of logical systems we can use to do so, they cannot be a complete ban on all proofs, since we can prove that the above assertion must be true.

Or can we? The reason it seems true is that right at the point where any such proof (of the impossibility of proving anything) is finally completed we realise that our proof invalidates the system of proof which we have used to prove it. Because of this I at first concluded that the assertion cannot be false and therefore must be true.

However it suddenly dawned on me that if it really is impossible to prove anything then the assertion above would still be true. It would be true even though we couldn’t ever prove it to be true. It would be just a simple statement of fact. There are two reasons why it might be true, one because logically it must be true, and secondly because it is true.

This may seem a trivial point but it isn't. It means that even though the assertion is true we still know nothing about whether it is possible to prove anything or not, since it would remain true whether or not it is impossible to prove anything.

We would normally take the self-contradictory nature of the assertion as meaning quite literally that ‘it is impossible to prove that it is impossible to prove anything’. We would claim that ‘intuitively’ we know that the assertion shows that no proof of the impossibility of proof is possible. But because there are two possible reasons for the assertion being true we still do not know whether or not we can ever prove anything.

(I'm doing my best!)

If we cannot prove that ‘it is impossible to prove anything’ then we cannot prove that ‘it is not impossible to prove anything’.

______________________

Let us say that

p = any complete proof of the impossibility of proving anything
and
s = any system of logic within which p may be constructed

If p is true then it is a complete proof, if it is false then it is not.
If s is true then it is logically consistent and can in theory produce a true p , if false then it cannot produce a true p.

Then

If p is provably true then s is provably false, in which case p is provably false, and therefore s is not provably false….and so on ad finitum.

OR

If p is true but not provably so then all s's are false.
__________________________

The unavoidable logical consequence of this strange situation is that the question of whether it is possible to prove that all proofs are impossible must be in principle undecidable. We might assume, as we did at the start, that one answer is more logical than the other, but this is just an assumption, there can be no proof of it. We have a choice. We can assume that the self-contradictory nature of the original assertion is proof that we cannot prove the impossibility of proof, or we can assume that it is just further evidence that we cannot prove anything.

Intuitively we choose the former conclusion, since the latter seems ridiculous. But we must accept that it is not provably ridiculous. It is just intuitively ridiculous. The fact is that we have not proved that it is impossible to prove that that proof is impossible. Rather we have proved that the question of whether we can or not is undecidable.

Note that this undecidability here does not mean simply that we do not know the answer to the question yet. It means that the question is absolutely and forever undecidable by proof, that any attempted proof of either answer must end up being self-contradictory or incomplete.

But this is very odd. If we cannot prove that it is impossible to produce a proof that all proofs are impossible then it follows that all proofs are suspect, since we then know that a disproof of them is in principle forever possible. If all proofs are suspect in this way then we cannot ever conclusively prove that any particular proof is true. In this case it is not possible to conclusively prove anything.

It follows that after all it is possible to prove that it is impossible to prove anything, for the undecidability of our original assertion is the proof of it.

This seems to force on us the conclusion that in theory we can know that it is impossible to conclusively prove anything, but at the same time also know that it is in principle impossible to ever prove it. That is to say that there is more to ‘knowing’ than than there is to proving. We can know things are true even while knowing that we cannot prove them within any formal system of logic.

Now this is simply a revised version of Goedel’s mathematical proof that in any complex and formal axiomatic system of proof there must be more that is true than can ever be proved true. Not only can we know truths that we cannot prove, but it seems from the above we must assume something in order to prove anything. This is the link to eastern philosphy, in that the common opinion of non-dual philosophers is that to assert anything as provably true is to cause self-contradiction and intellectual confusion, and from this they seems to be quite correct.

In summary, (at last!) my contention is that if we cannot prove that it is impossible to prove the impossibility of proof then proof is impossible. This conclusion is entailed by the fact that if you cannot prove that it is impossible that any particular proof is false then all proofs are inconclusive, and thus they are all false.

What do you think? Am I mad? Are there logical errors in the above reasoning? Is it possible to prove that it is impossible to prove the impossibility if proof or not? Is it possible to prove anything or not? What do the concepts of truth and falsity mean if there is never a full proof of which is which? Is all this trivial nonsense or profoundly important?

Of the answers so far I would consider Xenu's 'Mu' as the most accurate.

Must stop there, nurse has brought my medication.

 

[sparkle]

...impressed silence...

*headspin* Wow, after translating your text to my own language and reading it over and over in both it and English I am still confused and not sure I got you. I have two suggestions:

1. Could you start over (should I dive for cover here?) with the „inner“ hypothesis („it is impossible to prove anything“) and then, once you complete that move to the outer one (the possibility of proving the „inner statement“)? It would IMHO look like that:
it is impossible to prove anything - three options: true, false, neither.
Then IF true, the possibility to prove...
IF not true, the possibility to prove,
for neither, hm, I don’t know. Should be a conclusion in a sense. I noted you did that in a way, but it seems not structured enough to me. I had problems following you through the „inner“ and „outer“ statements.

2. Could you be a bit more specific? For me the way of thought was: It cannot be impossible to prove the impossibility of anything, because if there is a single incident where it would be possible, it is impossible to be impossible to prove the impossibility of anything *sigh* and therefore it would be possible to prove the impossiblity of something. *headsteam* Anyway, before it gets complicated again, I declare I will sacrifice one of my examples for you to rip to pieces. Deal?

 

[Canute]

Sparkle - Yes it's a nightmare trying to unpick it. I did my best but couldn't find a way of putting it simply. Respect for trying to read through it.

I'm away for a while so can't add anything now. I'll definitely do as you ask when I come back since I want some help getting to the bottom of it.

It helps if you've thought about Goedel, the Liar's Paradox and so on. (Btw discussions of this kind of self-reference always get very confusing, it's not just me).

Back soon.

 

[gendanken]

Lifer:

My point is that this admitance should mirror the demise of those particular philosophies. For how can we trust a religion that does not know certain things?

So you'd hedge bets on the side that insists that its right and goes out to fell empires and whole peoples because its convinced that its a-b-s-o-l-u-t-e-l-y right?

That's been the nasty habit of the Western mind. That's what distills from "absolutelies"- chaos. Surely you've noticed.........or you'd rather keep to your sophies instead?

 

[Canute]

Re: ...impressed silence...

Originally posted by sparkle
*headspin* Wow, after translating your text to my own language and reading it over and over in both it and English I am still confused and not sure I got you. I have two suggestions:

1.Could you start over (should I dive for cover here?) with the „inner“ hypothesis („it is impossible to prove anything“) and then, once you complete that move to the outer one (the possibility of proving the „inner statement“)? It would IMHO look like that:
it is impossible to prove anything - three options: true, false, neither.
Then IF true, the possibility to prove...
IF not true, the possibility to prove,
for neither, hm, I don’t know. Should be a conclusion in a sense. I noted you did that in a way, but it seems not structured enough to me. I had problems following you through the „inner“ and „outer“ statements.

2. Could you be a bit more specific? For me the way of thought was: It cannot be impossible to prove the impossibility of anything, because if there is a single incident where it would be possible, it is impossible to be impossible to prove the impossibility of anything *sigh* and therefore it would be possible to prove the impossiblity of something. *headsteam* Anyway, before it gets complicated again, I declare I will sacrifice one of my examples for you to rip to pieces. Deal?

OK, here goes.

The problem is concerned with the relationship between truth and proof. It is an attempt to reproduce Goedel's proof that they are not the same thing without all the mathematics.

To put it imprecisely Goedel proved that in any system of thinking capable of being translated into a formal axiomatic system of proofs, (ie all 'rational' ways of proving things about the world) there are things that are true which cannot be proved to be true, and that all systems of proof (and thus all proofs) are flawed. (Some might argue with this way of putting it but I'll be here all day if I try to justify this interpretation - I'll come back to it if you disagree).

Please note that this discussion applies to axiomatic systems only, systems that refer outside of themselves ('synthetic' systems in Kant's sense, rather than analytic systems, statements in which are simply tautological and thus completely decidable by proof).

In any system of proof it is possible to express the statement 'it is impossible to prove anything'. However in all systems of proof it is impossible to prove this statement to be true. This is for logical reasons.

One can always assert that 'it is impossible to prove that it is impossible to prove anything' and know that it is true. If it were false it would be self-contradictory, since if it were false then one could assert the opposite to be true, namely that 'it is possible to prove that it is impossible to prove anything'. This assertion is plainly nonsensical since if it is true it is false.

From this we might conclude that the assertion 'it is impossible to prove that it is impossible to prove anything' must be true. We might also therefore conclude that it IS possible to prove some things. (By 'prove' I mean completely and rationally prove).

But this is jumping the gun. There are two reasons that our assertion might be true. Let's call the assertion, which can be made within any rational system of thinking based on proof of truth and falsity you choose, A(S).

A(S) seems definitely true for the logical reasons given. Logically it cannot be false. Thus whether we can completely prove anything is an empirical matter. It is in principle impossible to prove otherwise, since to prove otherwise is to contradict ourselves.

But A(S) may also be true because it is actually TRUE. In other words, and despite the fact that we can never prove it, it may really be the case (as Goedel suggests) that we cannot prove anything completely by any rational method of proof. In other words A(S) may be just one example of what we cannot prove.

Thus although the assertion cannot be false (and therefore Goedel cannot have proved that proof is impossible) we still know nothing about whether it is possible to prove anything or not, since A(S) would remain true whether or not it is impossible to prove anything.

That is to say that although we cannot PROVE that proof is impossible it may yet be true that a complete proof of anything about the universe is impossible.

From Goedel we know that we can know things which we cannot prove. The question therefore arises of whether it is possible to KNOW that all proofs based on rational systems of truth and falsity must ultimately fail, without being able to prove it.

To update my example-

Let us say that

p = any complete proof of the impossibility of proving anything
and
s = any system of logic within which p may be constructed

If p is true then it is a complete proof, if it is false then it is not.
If s is true then it is logically consistent and can in theory produce a true p , if false it is inconsistent and cannot produce a true p.

Then

If p is provably true then s is provably false, (it must be inconsistent since p is logically self-contradictory). However if s is provably false then p is not provably true (since the system of proof is logically flawed). Thus p cannot ever be proved to be true.

BUT - there is another interpretation.

If p is true but not provably so then all s's are false. (This is the link with Eastern philosophies, most of which assert that all proof is based on dualism (truth and falsity) and thus that all knowledge derived from such proofs is suspect and subordinate to what we know from our own unprovable experience).

This leaves us in the position that IF it is the case that all complete proofs are impossible then we will never be able to prove it.

This means that when someone claims that 'it is impossible to completely prove anything' (and thus that all third-person proof-based systems of rational thinking are ultimately NOT capable of completely proving anything logically) it is impossible to refute them, since we cannot expect them to be able to prove it. The fact that they cannot prove it is part of their evidence.

(This is getting away from me but I'll keep going).

If a priori we know that we cannot refute their claim (since they can offer no proof of it which we can then disprove) then all proofs are ultimately suspect.

If all proofs are suspect in this way then it is true that it is impossible to prove anything, even though we cannot prove it.

Thus we have solved our problem in the same way as Goedel. he showed that it is possible to KNOW that a Goedel sentence is true although it is impossible prove it to be true (within the existing system of proof). In the same way it seems to be (theoretically) possible to KNOW that all rational proofs of anything (non-trivial) are impossible but be unable to prove it.

Phew! This went wrong again but I'll leave it as it is. I cannot clinch the argument because I'm arguing that all arguments are unclinchable. Anyway please argue with it because it might help sort out the confusion. There must be a simpler way of putting it but I can't find it.

 

[Canute]

RESULTS

The majority of respondents (10) voted the assertion false. I may have missed something but I cannot see how this can be the right answer.

7 people voted it true. This seems to me to be the correct answer.

3 people voted it undecidable. I feel that this is the wrong answer but the most interesting one. This is because although the assertion MUST (imo) be true it may be true for one of two very different reasons, and thus the reason why it is true is undecidable, which may have been what these three people noticed.

If the whole thing just confused you then welcome to the club.

 

[Voltaire]

I believe it would be impossible. Like you said true/false is all dualistic and all is based on your perception. I get your question, personally i don't think it is that complicated. I just keep in mind there are infinite possibilities and it would be extremely difficult to prove they all exist.

 

[Morteza Olangui]

Impossible has become possible!

Hi
Once upon a time, there were people on Earth that thought many things were impossible for them to do. But there were a few who thought other wise. With their hard effort they proved us that they were right.
Because of their painstaking endeavor, now, you and I can sit before our monitors and keyboards and get in touch with one another wiht just a mouse click. The impossible has become possible. Thanks

 

[Voltaire]

Re: Impossible has become possible!

Originally posted by Morteza Olangui
Hi
Once upon a time, there were people on Earth that thought many things were impossible for them to do. But there were a few who thought other wise. With their hard effort they proved us that they were right.
Because of their painstaking endeavor, now, you and I can sit before our monitors and keyboards and get in touch with one another wiht just a mouse click. The impossible has become possible. Thanks

huh? i don't think you get what i am saying, there are so many possibilities, theorems, and such that it would be imposible to prove them all. we didn't get monitors from one day to another. people need the time to evolutionize in their own given time. do u get what i mean?

 

[Morteza Olangui]

Did I...?

hi:
Dear voltaire:
Did I say these impossiblities-turned- into- commodities were made in a day? Very simply, our great great...grand parents once thought the whole world was created by God in six days. One day this one day that and at last, Dear Eve and Adam came into being. That was impossible for them to understand the multi-billiard old universe as you and I do now. Do you not think that some day this verbal lovemaking of philosophers will end and they will be satistied and say: Halelluya? Be my guest.

 

[Voltaire]

Re: Did I...?

Originally posted by Morteza Olangui
Do you not think that some day this verbal lovemaking of philosophers will end and they will be satistied

no. again i repeat for the hundredth time, there are so many things to prove that the inquiring nature of humans will never end and it would be impossible to prove them all.