Can We Prove Anything?

[durangodawood]

Can We Prove Anything in some kind of ultimate sense?

I think its not possible in math, as demonstrated by Godel, the overall system being either inconsistent or incomplete. So proofs are kind of conditional. (?) But perhaps his proof about formal systems hits 'rock bottom'. (?) Anyway.....

Is it possible in any other realm?

 

[ToddNotTodd]

Can We Prove Anything in some kind of ultimate sense?

I can prove to myself that at least something exists.

 

[durangodawood]

I can prove to myself that at least something exists.

Fair enough.
I guess there's two ways of looking at proof.
1. something you can communicate, demonstrate
2. something utterly private, that you cant show to anybody.
I'm more interested in the 1st. (Tho your example is spot on, imo)

 

[essentialsaltes]

Can We Prove Anything in some kind of ultimate sense?

I think its not possible in math, as demonstrated by Godel, the overall system being either inconsistent or incomplete.

Gödel showed that a single mathematical system can't prove every true statement. But it doesn't prevent us from proving some true statement. For instance, the Pythagorean Theorem has been proven in Euclidean Geometry.

 

[durangodawood]

Gödel showed that a single mathematical system can't prove every true statement. But it doesn't prevent us from proving some true statement. For instance, the Pythagorean Theorem has been proven in Euclidean Geometry.

Thanks. I was hoping someone would set me straight.
Anything else outside of math you think can be proven?

 

[essentialsaltes]

Thanks. I was hoping someone would set me straight.
Anything else outside of math you think can be proven?

Similar to the assumptions of Euclidean geometry, if we accept the axioms of some particular system of logic, things can be proven.

All men are mortal.
Socrates is a man.
Socrates is mortal.

For 100% absolute certainty, I think math and logic are it. (And even then they rest upon particular assumptions.) Everything else is more like 'beyond a reasonable doubt', like the generally spherical shape of the earth. (Some unreasonable people here may disagree.)

 

[durangodawood]

Similar to the assumptions of Euclidean geometry, if we accept the axioms of some particular system of logic, things can be proven.

All men are mortal.
Socrates is a man.
Socrates is mortal.

For 100% absolute certainty, I think math and logic are it. (And even then they rest upon particular assumptions.) Everything else is more like 'beyond a reasonable doubt', like the generally spherical shape of the earth. (Some unreasonable people here may disagree.)

Your logic example relies on an unprovable premise.
As for math, the very notion of axioms means that some things are just assumed true, and not provable. Is that right?

 

[essentialsaltes]

Yes, we have to make some axioms. That defines a system. Then within the system, statements either are or aren't provable.

The Pythagorean Theorem is provable in Euclidean geometry.

The Pythagorean Theorem is not provable (indeed false) in non-Euclidean geometry.

All proof is tied to a system. And all systems will require some basic axioms.

If you have no axioms, you don't even have a way of defining what a proof is.

 

[GrowingSmaller]

I dont think that reality is designed, in terms of the reality of our cognitive life etc to be a system of proofs.

We work with a stable-ish system of governement, and theres also room for contingencies etc.

The sun rises, but the bird song is slightly different each morning....

So the brain (or soul, with its emotional and intellectual levels) works primarily with a non-linear reality, it gravitates towards normality but i a way which is unpredictable at the level of fine detail. Aha a rain drop.

Hats off to maths, but I just heard a thunder clap! Proof is not needed there. We evolved in different circumstances. Or, we are designed to deal with reality, not paperwork and symbolisms.

I think that to prove "real" things, we would need a theory of everything which could be used to derive every historical event, large or miniscule. Superstrings gone wild. And then a way of proving or rephrasing it such that its logic was seen to be coherent etc.

"Ibn `Abbas said: The first thing God created was the Preserved Tablet, on which was preserved all that has been and ever shall be until the Day of Resurrection. What is contained thereon no one knows but God. It is made of white Pearl"

 

[Chriliman]

Can We Prove Anything in some kind of ultimate sense?

I think its not possible in math, as demonstrated by Godel, the overall system being either inconsistent or incomplete. So proofs are kind of conditional. (?) But perhaps his proof about formal systems hits 'rock bottom'. (?) Anyway.....

Is it possible in any other realm?

Whatever is true will prove itself, it's up to me to accept it as true.

 

[ananda]

Can We Prove Anything in some kind of ultimate sense?

I think its not possible in math, as demonstrated by Godel, the overall system being either inconsistent or incomplete. So proofs are kind of conditional. (?) But perhaps his proof about formal systems hits 'rock bottom'. (?) Anyway.....

Is it possible in any other realm?

The only things we can truly prove are the things we have proven to ourselves. All others are only teachers and guides along the way.

 

[quatona]

Can We Prove Anything in some kind of ultimate sense?

I think its not possible in math, as demonstrated by Godel, the overall system being either inconsistent or incomplete. So proofs are kind of conditional. (?) But perhaps his proof about formal systems hits 'rock bottom'. (?) Anyway.....

Is it possible in any other realm?

No, we always need to start from axioms. Interestingly, though, the question "Can we prove anything?" itself is built on some axioms.

 

[GrowingSmaller]

The word "prove" is ambiguous. Sometimes it just means "show"", as in "prove that its sunny".

 

[durangodawood]

The word "prove" is ambiguous. Sometimes it just means "show"", as in "prove that its sunny".

Good point. I intend it as the strictest version of "prove", to which there are no alternatives, like within mathematics.

 

[durangodawood]

No, we always need to start from axioms. Interestingly, though, the question "Can we prove anything?" itself is built on some axioms.

I agree with this.
So there is no rock-bottom proof of anything. We have to start with some agreed assumptions.

 

[quatona]

I agree with this.
So there is no rock-bottom proof of anything. We have to start with some agreed assumptions.

Yes, but my point was a different one: This is not only a problem of "proof", it´s already a problem with any simple statement or question.
Considering this, it´s not that much of a problem because some "assumptions" are obviously inevitable (and therefore implicitly agreed upon).

 

[durangodawood]

Yes, but my point was a different one: This is not only a problem of "proof", it´s already a problem with any simple statement or question.
Considering this, it´s not that much of a problem because some "assumptions" are obviously inevitable (and therefore implicitly agreed upon).

There's huge disagreement over whether 'miracles' are possible. Once you split on those lines, there's no common axiomatic basis for a lot of rather basic things.

 

[quatona]

There's huge disagreement over whether 'miracles' are possible.

Indeed.

Once you split on those lines, there's no common axiomatic basis for a lot of rather basic things.

Well, since "miracle" is defined as something pretty unusual I wouldn´t conclude from "miracles" on "rather basic things".

 

[GrowingSmaller]

Well there are proofs in math and logic. If you deny that you're stipulating a new definition of "proof", etc.

 

[durangodawood]

Indeed.

Well, since "miracle" is defined as something pretty unusual I wouldn´t conclude from "miracles" on "rather basic things".

But who knows what unusual event was a miracle? The end event of the dinosaurs? The big bang?