Can you proof proof?

[questionposter]

How do you prove that something proves something in number theory? Or how do you prove that something is proven?

Edited March 20, 2012 by questionposter

[Bignose]

You start with some axioms which are assumed to be true and derive logical consequences because the axioms are taken to be true.

[questionposter]

You start with some axioms which are assumed to be true and derive logical consequences because the axioms are taken to be true.

 

But you have to "assume" they are true. Is there any way you can actually prove those assumptions that something proves something?

[Bignose]

You gotta start somewhere. Axioms, by definition, are not meant to be provable. They are taken to be true. Without them, what can you even do? Axioms are obviously driven by things we observe and find useful. But, there is no proving them. If they could be proven, they would be based on other axioms, and wouldn't be axioms themselves. Like I said, gotta start somewhere.

[khaled]

There are two approaches, One is that you want to prove that something is valid .. so you start from mathematics axioms (things we

 

know are certain true) and try to derive logically that something.

 

Just like what Bignose said "Axioms, by definition, are not meant to be provable. They are taken to be true.",

 

axioms of mathematics were considered as a unified system, such that there wouldn't exist P and not P at the same time,

 

that's part of history, mathematicians who worked for unifying all mathematics like David Hilbert and others.

 

The other one is that you want to prove that two things can form a valid model, in which one can be derived from the other,

 

and they don't contradict, in other words, it has to be valid and consistent.

 

And to make sure that it's valid, you try logical derivation in both directions ...

Edited March 28, 2012 by khaled

[doG]

But you have to "assume" they are true. Is there any way you can actually prove those assumptions that something proves something?

No. An axiom is a proposition regarded as self-evidently true without proof, also called a postulate. Consider for example Euclid's first postulate, a straight line segment can be drawn joining any two points.

[questionposter]

No. An axiom is a proposition regarded as self-evidently true without proof, also called a postulate. Consider for example Euclid's first postulate, a straight line segment can be drawn joining any two points.

 

But that isn't true, we can observe that because of the warping of the fabric of space that that isn't true. In fact, it's never true because wherever your trying to draw a straight line, there is gravity distorting the fabric of space.

[John]

But that isn't true, we can observe that because of the warping of the fabric of space that that isn't true. In fact, it's never true because wherever your trying to draw a straight line, there is gravity distorting the fabric of space.

 

Euclid probably wasn't aware of the curvature of spacetime.

 

There are others geometries we can use to deal with such things, though Euclidean geometry is still fairly useful, of course.

 

You might want to read through the Wikipedia article on "axiom."

Edited March 29, 2012 by John

[Ben Banana]

Neither does Euclid require that awareness.

Non Ultimum Fundamentum

 

Edit:

I'm bothered. Please say: "Can you prove proof?" next time.

----

 

http://en.wiktionary.org/wiki/metaphysics

This common definition made is etymologically skewed.

Edited March 29, 2012 by Ben Bowen

[khaled]

Euclid probably wasn't aware of the curvature of spacetime.

 

Imagine we are at the time, when Euclid or Archimedes was living .. We wouldn't be aware of much

[kavlas]

You start with some axioms which are assumed to be true and derive logical consequences because the axioms are taken to be true.

 

 

How do you define logical consequence?

[Bignose]

How do you define logical consequence?

 

statements that can be demonstrated true, given the current axioms.

[kavlas]

statements that can be demonstrated true, given the current axioms.

 

And how can statements be demonstrated true??

[Bignose]

And how can statements be demonstrated true??

 

because they follow based on the axioms taken to be true. You may want to look into some of the documents on Introduction to Proof, or Introduction to Logic or similar books/webpages.

[Greg H.]

But that isn't true, we can observe that because of the warping of the fabric of space that that isn't true. In fact, it's never true because wherever your trying to draw a straight line, there is gravity distorting the fabric of space.

 

 

And in a curved space-time plane, the Euclidean axioms would not hold true, but there would be other axioms used in their place. An axiom is only useful (and accepted as true) in the framework it's defined in.

 

Some other Euclidean postulates (from Wikipedia):

• A straight line may be extended to any finite length.
• A circle may be described with any given point as its center and any distance as its radius.
• All right angles are congruent.

I'm sure some of these would also fall by the way side in a non-Euclidean space, but when you're dealing with flat surfaces they all hold true.

[kavlas]

because they follow based on the axioms taken to be true. You may want to look into some of the documents on Introduction to Proof, or Introduction to Logic or similar books/webpages.

 

I did not ask why ,but how can statements along a proof be demonstrated to be true.

 

Any way thanks for the help so far.

 

I did a google journey but it was not very satisfactory. Everything is so obscure and not very clear w.r.t the mechanisms of a proof.

 

I wander is it so difficult to really analyse a mathematical proof??

 

I also wander what are the constituents of a mathematical proof