```Date: Apr 6, 2013 6:34 PM
Author: fom
Subject: Re: Matheology § 224

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 theoriesthat constitute their deductive closure.If, however, you wish to apply that freedom to anestablished theory, other responsibilities arise.First of all, it will be a different theory.  Toclaim that it is a corrected theory is to make aphilosophical argument that the difference betweenthe original theory and the corrected theory reflectssome typical expectation or standard practice ofmathematicians -- outside of foundations -- thathas not been represented in the original theory.Second, if one is not claiming that it is acorrected theory, then one must be clear thatit is not the standard theory.  If it is notthe standard theory, then the relation to thestandard theory becomes an issue.  The relativityprinciple you espouse must be shown to beapplicable.  It can only be applicable if itdoes not interfere with the possibility of aproof within the standard theory.  Thiscriterion of applicability translates intoa restriction of relativity to those statementswhich have been shown to be independent.Statements are shown to be independent byformulating a model in which the statementis true and formulating a model in whichthe statement is false.There is nothing in these remarks that doesnot respect, first and foremost, the possibilityof a syntactic proof of an unproven statementwithin the standard, established axioms.
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