In my mathematical work which involves testing model graphs as hypotheses in evolved, recurrent neural networks, Gödel's first theorem states that there may be true models that cannot be proven as true in a formal axiomatic system. Thus, "truth" is an underdetermined state when it comes to the application of enumerable axiomatic properties in this arithmetic formal system.
When we talk about true and false, we are really talking about two coupled systems 1) false - not false, with 2) true - not true. The intersection yields for "senses"[Dominic Widdows - A Mathematical Model of Word-Sense Disambiguation] of true and false - being TRUE, FALSE, UNDETERMINED, and PARADOX. Gödel's second theorem states that a formal axiomatic system is complete if and only if it is inconsistent. The tack I take is I will go for consistency over completeness any day. But that's just me, and which probably disqualifies me as a philosopher in any canonical, categorically imperative sense. I just Kant do it. Ken > -----Original Message----- > From: [EMAIL PROTECTED] > [mailto:[EMAIL PROTECTED] On Behalf Of Owen Densmore > Sent: Thursday, July 17, 2008 10:41 AM > To: The Friday Morning Applied Complexity Coffee Group > Subject: Re: [FRIAM] Confessions of a Mathemechanic. > > On Jul 17, 2008, at 10:27 AM, Roger Critchlow wrote: > > On Wed, Jul 16, 2008 at 8:57 PM, Owen Densmore <[EMAIL PROTECTED]> > > wrote: > >> No one who accepts mathematics as it is, however, considers it a > >> point of philosophy. We do not argue about it, we try to grasp it. > >> > >> Arguing about it is for those of us who cannot understand it. > >> > > I suspect a category error: was Goedel's theorem mathematics or an > > argument about mathematics? > > The former. > > I do admit Gödel creates an interesting problem (not argument) for > mathematicians: You MUST be careful about your axioms, and > you should be aware of the problems they present. > > My hazy understanding of Gödel's work is that basically an > axiom set can be over-specified (thus creating the potential > for both T and !T being provable) or under-specified (T is > true but not provable). This is old stuff for linear algebraists. > > All that said, how many mathematicians are halted in their > tracks by Gödel, giving up all as foolish and pointless? > Rather they use it as a cautionary tale, much like computer > scientists dealing with decidability. > > -- Owen > > > ============================================================ > FRIAM Applied Complexity Group listserv > Meets Fridays 9a-11:30 at cafe at St. John's College > lectures, archives, unsubscribe, maps at http://www.friam.org ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org
