As for the Godel stuff, there's a part of it with which I disagree (or at least as far as I take what you said).
If you want to compare something mathematically you -must- use the same axioms and rules of derivation. The -only- discussion there is one of two parts: Is the sequence of applications/operators valid? (ie Proof) Is the sequence terminal, does it leave room for more derivation? (ie Publish or Perish)
Well, not necessarily, unless I misunderstand you. Take the Fermat's last theorem example I gave (a^n+b^n=c^n for a,b,c,n integers but n>2).
And let's say I want to "prove" (or disprove) the statement "This has no solution for n>2.
There are two 'distinct' methods of determining the validity of the statement. One is by what is normally considered a "proof". In other words, by building up from axioms using the logical rules of the system.
The other is to actually find a solution for a,b,c and n. In this case the statement will have been disproven, but not by a series of logical statements and axioms. It is now seen to be "untrue", but not via the methods of "proof". Thus, the statement is untrue, and (possibly) unprovably untrue (which is the same thing as saying the statement's negation is "unprovably true").
Now if subsequent truths need to be made but require the statement above (a^n+b^n=c^n has no solution for n>2), even though we know that it is true (or untrue, in my example above), to build subsequent truths we need to include this statement as an axiom even though we know it's true. It's "true", but unprovable.
But perhaps this is what you meant.
Yes, I agree, and the confusion to which I referred had to do with the term "true" as it seemed to be used by various parties in the conversation. From this alone I think a big "take away" here is that "true" in the Godelian sense means something probably quite different from what many believe it to be.And no, there is zero confusion on what true means under Godel or Cauchy.
Couldn't agree more. "Reason is the whore of desire." Well, not always, but its clear to me that most of the time we start with the conclusion we want and then work backwards! Most human beings seem to stumble upon some little piece of flotsam and then cling onto it for dear life, not knowing they can actually swim (or perhaps they don't need to!). I don't consider myself an exception, except for the fact that knowing this, I constantly try to expose myself to information and experiences that do not correspond to what I currently believe. As the spanish mystic St John of the Cross wrote:The reality is that most people have problems grasping concepts or ideas because there is a conflict with other ideas/concepts they hold dear and near. In most cases of mental block it is an emotional issue not an intellectual one. People have a hard time learning not because they are stupid but because they don't deal with their emotional landscape effectively.
"To come to be what you are not, you must go by a way in which you are not. To come to know what you know not, you must go by a way in which you know not."
_________________________________________________________________
Tired of spam? Get advanced junk mail protection with MSN 8. http://join.msn.com/?page=features/junkmail
