On 4/6/2013 4:56 PM, Nam Nguyen wrote: > On 06/04/2013 2:34 PM, fom wrote: >> On 4/6/2013 11:48 AM, Nam Nguyen wrote: >>> On 05/04/2013 12:20 AM, fom wrote: >>>> On 4/5/2013 12:57 AM, Nam Nguyen wrote: >>>>> On 04/04/2013 10:55 PM, fom wrote: >>>>>> >>>>>> Who knows what is and what is not -- even >>>>>> in the simple realm of mathematics -- claims >>>>>> a certain knowledge that is revealed rather >>>>>> than discerned. >>>>> >>>>> So, since Godel, is the knowledge of the natural numbers >>>>> a revealed or discerned one? >>>>> >>>>> Revealed by whom? Discerned from what? >>>>> >>>> >>>> I thought you claimed to be a relativist. ??? >>> >>> I am, by at least the 3rd Principle "Principle of Symmetry (of >>> Non-Logicality)" mentioned in: >>> >>> http://groups.google.com/group/sci.math/msg/20bb0e7c183ae502?hl=en >>> >> >> What appears to be a problem with your principle is >> that one does not know what is and what is not provable >> to begin with. > > You seem to misinterpret the principle, which is actually > a logical one. > > Assuming that the formula A is neither a tautology or contradiction, > it's impossible to conclude A or to conclude ~A from (A \/ ~A). > Therefore it's _relative_ to your choice to choose which of A, or ~A > be your axiom. Ditto for the dichotomy (A \/ B): it's relative to > which of A and B you'd choose. This is in the realm of syntactical proof > via rules of inference.
I have not misinterpreted your principle.
You are free to construct axioms and the theories that constitute their deductive closure.
If, however, you wish to apply that freedom to an established theory, other responsibilities arise.
First of all, it will be a different theory. To claim that it is a corrected theory is to make a philosophical argument that the difference between the original theory and the corrected theory reflects some typical expectation or standard practice of mathematicians -- outside of foundations -- that has not been represented in the original theory.
Second, if one is not claiming that it is a corrected theory, then one must be clear that it is not the standard theory. If it is not the standard theory, then the relation to the standard theory becomes an issue. The relativity principle you espouse must be shown to be applicable. It can only be applicable if it does not interfere with the possibility of a proof within the standard theory. This criterion of applicability translates into a restriction of relativity to those statements which have been shown to be independent.
Statements are shown to be independent by formulating a model in which the statement is true and formulating a model in which the statement is false.
There is nothing in these remarks that does not respect, first and foremost, the possibility of a syntactic proof of an unproven statement within the standard, established axioms.