Not everything that is true can be proven

Deleted member 115935

Joined Dec 31, 1969
0
Scientific theories are built from empirical hypotheses, not formal statements, and so can neither be proved nor disproved. At best, we can continue to accept or reject the hypotheses based on the accumulation of evidence.


What exactly do you mean by "hole"?


In many mathematical contexts, division by zero is simply undefined. It doesn't have to be -- we can define 0/0 axiomatically or as shorthand for a limit. The only restriction is that we don't introduce any inconsistencies.

By hole,
I mean things we can not prove, but are taken as "true" till proven otherwise,
a big bunch of them @bogosort, jenifer solomon says your a mathematician, Im not , just an engineer,
please, these problems are outside my areas, may be you could expand on them for us.
would be very interesting ,

https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics
 

bogosort

Joined Sep 24, 2011
696
By hole,
I mean things we can not prove, but are taken as "true" till proven otherwise,
It's difficult to parse exactly what you mean by that. The phrase "can not prove" may mean have not yet found a proof, or it may mean we know there can never be a proof. In any case, this is a deep rabbit hole. The core of the issue -- and the source of much confusion -- is that provability and truth are orthogonal concepts.

Provability belongs to the realm of syntax. It is a purely formal thing: given this language with these symbols, these axioms written in the symbols, and a set of rules of inference, can we derive the given statement? Provability comes from logical deduction, which is entirely mechanical.

Truth, on the other hand, is a semantic notion. When we apply meaning to the symbols in a formal theory, we give it semantics by associating the theorems with truth values. Such an application creates a model from the theory. Unfortunately, all of this is further complicated by the fact that there are different "orders" of theories, with different properties and relations. Suffice it to say that for second-order theories (which is what most math is tacitly using), a formal statement is true if and only if it is true in every model of the theory. That is, no matter what semantics we apply, a true 2nd-order statement will be true under any interpretation.

Everything gets even more snake-eating-its-tail confusing once we start using mathematics to formally describe mathematics itself, which is what Gödel did.
 

MrAl

Joined Jun 17, 2014
11,581
Everything that is true *CAN* be proven. Do not confuse our ability to prove something with it's ability to be proven. This is a typical 'man's folly' argument. Man attempts to view everything through his own limitations.
It gets very interesting when logic tries to prove itself, which happens when logic is contained within itself.
For example, can you prove that "Everything that is true CAN be proven"?

There is also the 'paradox'.
For example:
"In this town, the barber shaves everyone that does not shave themselves".
Now prove who saves the barber.

Then we get into the physical derivative...
Prove that a hole in the ground is not a physical thing in and of itself.
I think this is possible though.
 

Deleted member 115935

Joined Dec 31, 1969
0
is a hole in the ground a object, or just a label for a lack of ground.

but this the sort of stuff that the friends bogosort and jennifer solomon are great at talking on,
 

BobaMosfet

Joined Jul 1, 2009
2,119
A proof is ultimately a logical derivation of a true statement within some formal (axiomatic) system. Without the formal system, which defines the language and rules for making such derivations, there is no proof. In other words, a proof is only valid with respect to some given formal system.

Gödel showed that for any such system, if it is powerful enough to express arithmetic over the integers, then it necessarily contains theorems (true statements) that are unprovable. The demonstration of this takes some work, but it is indisputable.

This is not a folly of man; it is a fundamental limitation of any given formal system. But without a formal system, you can't prove anything. So whatever you are talking about, it's not about actual proofs.
You've missed the point of my statement.
 

BobaMosfet

Joined Jul 1, 2009
2,119
It gets very interesting when logic tries to prove itself, which happens when logic is contained within itself.
For example, can you prove that "Everything that is true CAN be proven"?

There is also the 'paradox'.
For example:
"In this town, the barber shaves everyone that does not shave themselves".
Now prove who saves the barber.

Then we get into the physical derivative...
Prove that a hole in the ground is not a physical thing in and of itself.
I think this is possible though.
Everything that is true can be proven if you confirm that truth exists in all circumstances. An old axiom is: Take everything to its logical extreme to prove it. That is fundamental, and obvious. The problem with mankind is that he (typically) fails to maintain pure objectivity- he either fits the data to the problem, or he alters the problem to fit the data. Neither of which is necessary if what is true, is actually true.

Saying that one cannot prove what is true is simply a self-limiting prophecy by those who aren't really all that intelligent- hence why man continues to struggle with things that aren't all that hard.
 
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bogosort

Joined Sep 24, 2011
696
Saying that one cannot prove what is true is simply a self-limiting prophecy by those who aren't really all that intelligent- hence why man continues to struggle with things that aren't all that hard.
Is it true or false that every even integer greater than 2 is a sum of two primes? Prove it.
 

BobaMosfet

Joined Jul 1, 2009
2,119
Is it true or false that every even integer greater than 2 is a sum of two primes? Prove it.
Let's simplify:

Is it TRUE that every EVEN INTEGER greater than 2 is a SUM OF TWO PRIMES?

The answer is FALSE.

8 is an EVEN INTEGER, greater than 2, that is NOT a sum of TWO PRIMES. In fact, it is the SUM of TWO COMPOSITES.
 

bogosort

Joined Sep 24, 2011
696
I'll put it in terms you can (hopefully) understand. You can't say you can't prove something because you simply don't know how to prove it.
Thanks for dumbing it down for me. However, Godel didn't say that we don't know how to prove some statements of arithmetic (we don't need a theorem for that obvious fact). What he did was prove that, for any theory of arithmetic T, there are true statements in T that cannot be proved. How do we know that these are true statements? Because they are true if and only if T is a consistent theory. And since no theory can prove its own consistency (the second incompleteness theorem), then there must be unprovable true statements.

That is, if P is one of these true statements, then a proof of P is a proof that T is consistent. But this leads to a contradiction, as only inconsistent theories of arithmetic can prove their own consistency. Therefore, no such proof of P can exist within T.
 

MrAl

Joined Jun 17, 2014
11,581
Everything that is true can be proven if you confirm that truth exists in all circumstances. An old axiom is: Take everything to its logical extreme to prove it. That is fundamental, and obvious. The problem with mankind is that he (typically) fails to maintain pure objectivity- he either fits the data to the problem, or he alters the problem to fit the data. Neither of which is necessary if what is true, is actually true.

Saying that one cannot prove what is true is simply a self-limiting prophecy by those who aren't really all that intelligent- hence why man continues to struggle with things that aren't all that hard.
Well maybe in common experience, but things are getting very strange lately due to ideas in quantum physics which is turning out to be as profound and game changing as Einsteins ideas.

The way i used to describe this in a sort of fictional way depending on what you believe in, is that logic belongs to the devil that's why the devil wins every argument. The non fictional view would be simply that logic can be deceiving, and so much so that we cant pin some things down for certain that we normally thing we could. For example, for Einstein's train and lightening bolts, he said that two observers might view the lightening strike(s) as occurring at different times. What he assumes though is, that something ACTUALLY happened. In quantum physics it's a little different because two people might actually experience a completely different reality, so what one person 'proves' to have happened another person could prove that it didnt happen, and they would both be right.
I guess i could try to find some articles about this. It gets pretty weird though. It's almost like nothing we have been taught so far is right all the time and for every occurrence. Particles being entangled, people becoming entangled with objects, it gets strange. Mr. Schrodinger's cat was just the tip of the iceberg ... if you become entangled with the cat and you are in a box with the cat in a box then you cant relay that information to an outside observer even if you open the box that the cat is in.
But i guess the real kicker is that two people can observe and MEASURE something like spin and both measure something different, and they would BOTH be right. Maybe that is still along the lines of being able to prove something though because they can both prove it :)

But the incompleteness 'oracle' idea is very interesting in that we can create a falsity from all truths it seems. Maybe that's not quite 'not' the same as proving something though.
 

BobaMosfet

Joined Jul 1, 2009
2,119
Erm, 3 + 5?
As they say, haste makes waste. I replied hastily, and got wasted..... go you! Goldbach's conjecture of 1742 is fascinating and is likely provable or disprovable not by equation but simply through proof that the algorithm of the equation itself will work infinitely (or won't). How problems are solved is often accomplished by altering the perspective upon which the problem is viewed- this is what geniuses do. They have this remarkable ability to view things differently than anybody else.

More info for anyone interested: https://en.wikipedia.org/wiki/Goldbach's_conjecture
 
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BobaMosfet

Joined Jul 1, 2009
2,119
Well maybe in common experience, but things are getting very strange lately due to ideas in quantum physics which is turning out to be as profound and game changing as Einsteins ideas.

The way i used to describe this in a sort of fictional way depending on what you believe in, is that logic belongs to the devil that's why the devil wins every argument. The non fictional view would be simply that logic can be deceiving, and so much so that we cant pin some things down for certain that we normally thing we could. For example, for Einstein's train and lightening bolts, he said that two observers might view the lightening strike(s) as occurring at different times. What he assumes though is, that something ACTUALLY happened. In quantum physics it's a little different because two people might actually experience a completely different reality, so what one person 'proves' to have happened another person could prove that it didnt happen, and they would both be right.
I guess i could try to find some articles about this. It gets pretty weird though. It's almost like nothing we have been taught so far is right all the time and for every occurrence. Particles being entangled, people becoming entangled with objects, it gets strange. Mr. Schrodinger's cat was just the tip of the iceberg ... if you become entangled with the cat and you are in a box with the cat in a box then you cant relay that information to an outside observer even if you open the box that the cat is in.
But i guess the real kicker is that two people can observe and MEASURE something like spin and both measure something different, and they would BOTH be right. Maybe that is still along the lines of being able to prove something though because they can both prove it :)

But the incompleteness 'oracle' idea is very interesting in that we can create a falsity from all truths it seems. Maybe that's not quite 'not' the same as proving something though.
Actually things are not getting all that strange. The 'strange' aspect is due to the theoretical (meaning, non-factual, just plain old guess-work) "understanding" of some aspect of physics. It is simply undeniable that if something is true in every possibility of its existence, then it is universally true. Anybody who cannot understand this is bereft of basic ability to correctly conceptualize. When you begin theorizing endlessly, and then building logic upon theory (which is largely what any 'theoretical' math, physics or astrophysics field is about- theory)- you lose grip with reality if you go too far along that path.
 

BobaMosfet

Joined Jul 1, 2009
2,119
Thanks for dumbing it down for me. However, Godel didn't say that we don't know how to prove some statements of arithmetic (we don't need a theorem for that obvious fact). What he did was prove that, for any theory of arithmetic T, there are true statements in T that cannot be proved. How do we know that these are true statements? Because they are true if and only if T is a consistent theory. And since no theory can prove its own consistency (the second incompleteness theorem), then there must be unprovable true statements.

That is, if P is one of these true statements, then a proof of P is a proof that T is consistent. But this leads to a contradiction, as only inconsistent theories of arithmetic can prove their own consistency. Therefore, no such proof of P can exist within T.
This is fundamentally why you do not prove a theory with itself, you prove it externally by evidence that it holds true in all circumstances. Simple.
 

bogosort

Joined Sep 24, 2011
696
This is fundamentally why you do not prove a theory with itself, you prove it externally by evidence that it holds true in all circumstances. Simple.
BobTPH is correct. You're using the words "proof" and "theory" in some loose, non-standard way, certainly not in the way that Godel used them. FYI: you're criticizing an idea without understanding it and calling the "refutation" of its strawman simple.
 

BobaMosfet

Joined Jul 1, 2009
2,119
BobTPH is correct. You're using the words "proof" and "theory" in some loose, non-standard way, certainly not in the way that Godel used them. FYI: you're criticizing an idea without understanding it and calling the "refutation" of its strawman simple.
I'm simply going to back up- I reread post #22 which you wrote, and I like it.
 
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xox

Joined Sep 8, 2017
838
Conversely, not everything that can be proven is necessarily true. Radio-carbon date a rock only to find out that it had been sitting next to a strong neutron source the whole time. We've proven that it dates to era X, but forgotten to ask why. Things aren't usually so convoluted of course, but nevertheless something to keep in mind when considering "true" statements.
 
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