Axioms

[MishMish]

How does one determine what is "axiomatic" and what its validity is

[Skye]

By agreement.

[Sayonara]

That's why we only have 4 of them.

 

j/k

[aommaster]

ok, one thing, what does 'axiomatic' mean, scientifically?

[wolfson]

Of, relating to, or resembling an axiom

[aommaster]

what is an axiom?

[wolfson]

An established rule, principle, or law.

[aommaster]

ok. Thanx!

[YT2095]

a Hinge/Pivotal point, something to based upon.

 

I assume taken from the word "Axis"

[Dov]

Etymology:

Latin axioma, from Greek axiOma, literally, something worthy, from axioun to think worthy, from axios worth, worthy;

akin to Greek agein to weigh, drive -- more at AGENT

 

1) a maxim widely accepted on its intrinsic merit

2) a statement accepted as true as the basis for argument or inference :

postulate

3) an established rule or principle or a self-evident truth

[Tom Mattson]

An axiom of a theory is something that is not proven, but rather taken as self evident. Every theory has to have axioms because, due to the finiteness of our language and concepts, any theorist will eventually run out of words and ideas to define his axioms in terms of. A synonym is "postulate", such as the postulates of Euclidean geometry.

 

Axioms must also be independent to be considered "axiomatic". That is, it must not be possible to derive an axiom from among the set of axioms. There was a time when challengers of Euclid charged that his 5th postulate was not really a postulate, but actually derivable from the others. The charge was eventually found to be false, however.

 

Another interesting point is the work of Goedel, who proved that every formal system that is at least as powerful as arithmetic is either incomplete or inconsistent. That is, it either contains true statements that can't be proven from within the systems (incomplete) or, for some statement X, it is possible to derive both X and ~X from within the system. Euclidean geometry happens to be an example of a formal system that is both consistent and complete.

[MishMish]

Thanks Tom

 

Realized I had not expressed my question very well, but you got at it

 

I look at it in much more basic terms, that one can not get outside the full system to find some outside objective standard. At some point, one is simply going to have to say something is, by defiition or by calling it axiomatic (and I may be using the term wrong here,) otherwise at some point everything will eventually be self-referential

 

My question is how one determines what validity that has, or how one knows one has reached that point beyond which there are no further assumptions to make (how I have always defined the independence of axioms, perhaps that is incorrect?)

 

I can not find one, it all looks quite circular to me (my primary problem with logic I think)

 

Also occured to me after posting perhaps this really belongs in the philosophy section, but be that as it may

[Tom Mattson]

My question is how one determines what validity that has' date=' or how one knows one has reached that point beyond which there are no further assumptions to make (how I have always defined the independence of axioms, perhaps that is incorrect?)

[/quote']

 

There are two answers to this: One for systems that correspond to things that are known a priori, and one for systems that correspond to things that are known a posteriori.

 

First, a priori systems. These are systems such as math and symbolic logic. They are systems whose axioms are true by definition. For instance, one doesn't "prove" that two parallel lines never intersect, it is simply one of the axioms of Euclidean geometry. In a priori systems, you can never know that your list of axioms is finished, because you can always enlarge your system to contain more and more axioms. But, care must be taken to ensure that all the statements in the list are axioms. Else, you could be calling things "axiomatic" that can actually be derived from other axioms, in which case the burden falls upon you to prove that statement and elevate it to a theorem.

 

Second a posteriori systems. These are systems found in such areas theoretical physics. Their axioms are asserted, but cannot be accepted as true until either they or their logical (usually mathematical) consequences have been verified by experiment. That is, in science there is a way to verify axioms independently (albeit not in the absolute sense, but by induction) of the formal system used in the theory. You know you have enough axioms when you can derive statements that can describe all past observations, and that predict new results.

 

Also occured to me after posting perhaps this really belongs in the philosophy section, but be that as it may

 

Yeah, either that or Math, because Goedel is taught in both departments.

[MishMish]

Tom, with a bit of snipping:

 

"One for systems that correspond to things that are known a priori, and one for systems that correspond to things that are known a posteriori."

 

"That is, in science there is a way to verify axioms independently (albeit not in the absolute sense, but by induction) of the formal system used in the theory. You know you have enough axioms when you can derive statements that can describe all past observations, and that predict new results."

 

I am not getting to this board much at all lately, and am going to have to think on this

 

That last part is how I have described the criteria for accepting some model as the preferred, however

 

Someone once told me I seem to have a problem with induction, and he is probably right as I have no idea what he meant, and I have never been able to keep the distinction clear no matter how many times I have tried. As I said, to me logic is quite circular

 

Even the notion that two parralel lines do not intersect seems to me must have had an observational basis (though obviously limited) leading to the abstracted principle

 

The question has been brewing for other reasons, but the immediate impetus was two basic principles I accept, cause and effect and that for two objects to interact there has to be some common element by which that interaction can be effected

 

They seem obvious to me, yet I can find no justification

 

Don't mean to pester, and appreciate your help on it, feeling too I have missed something somewhere however...