Glen, You say math can jump in and out of context with 'meta-math', "a mechanistic method for "jumping out" of the context of any given mechanism into its entailing context." If you have a complete mathematical representation of a button, how would you derive a representation of a button hole from it?
Phil > -----Original Message----- > From: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED] On > Behalf Of glen e. p. ropella > Sent: Monday, August 11, 2008 5:33 PM > To: The Friday Morning Applied Complexity Coffee Group > Subject: Re: [FRIAM] Rosen, Life Itself > > Günther Greindl wrote: > > this: > > > > > The existence proof I'm pointing out as an example of how math is > more > > > than formal systems (Tarski's indefinability or the GIT) merely > shows > > > that what we call math is not fully captured by formal systems (or > _a_ > > > > is not true; interpreting Gödel/Tarski or whomever does not show > > anything of the kind without a _prior_ philosophy of mathematics > which > > is actually just begging the question. > > Hmmm. So, let's just examine the GIT. What is shown is that, through > a > math technique (Goedel numbering), it can be shown that any particular > (complex enough) formal system will either allow sentences that are > undecidable or that can be both valid ("true") and invalid ("false"). > > It seems quite clear that because we can use math to demonstrate that > formal systems are inadequate to the task of capturing all of math > means > that math is more than formal systems. > > I don't believe this requires a prior philosophy of math. All it > requires is the mechanical rigor of formal systems plus a method for > counting the sentences in a formal system. It seems to me that > mindless > inference could infer this (which is just another way of saying that I > believe the proofs of the GIT that I've seen ... a.k.a. I believe the > GIT is true ... ;-), with no disrespect intended towards Tarski or > Goedel. > > > In fact, as Webb argues, Gödel's (and Tarski is a "weaker" Gödel) > Inc. > > Theorem actually speaks _for_ mechanism (but this remark is > definitely > > too cryptic in this short email). > > I agree that the GIT argues for mechanism (as Rosen defines it), with > the exception of a mechanistic method for "jumping out" of the context > of any given mechanism into its entailing context. Meta-math is > precisely a methodology of "jumping in and out of the context" of any > given formalism. And, despite its name, meta-math is just math. > > Indeed, that's my point. Math handles this jumping in and out of any > particular context (witness category theory where the diagrams apply to > many different particular bodies of math) but formal systems does not > because formal systems only works with 1 alphabet, grammar, and set of > axioms at any given time. There's been no attempt (as far as I know), > within formal systems, to hop between formal systems in a rigorous way > ... to create measures/metrics of them with which to build up spaces of > them. Granted, there has been lots of discussion about the differences > between particular formal systems (e.g. ZFC and its variants). And, > also granted, there's been lots of work to demonstrate the equivalence > of various particular formal systems. > > But I don't know of any attempts to build a general theory that works > with all formal systems and their relationships. (I'd love it if > someone > would enlighten me to such efforts!) And even if there were, as we > regularize that theory, it will also be subject to the GIT, which means > we will again use math to study the limits of that system. > > No matter how high or low in the hierarchy you may go, you will still > be > using math, but you will not be locked within any given formal system. > Hence, math is somehow more than (or outside of) formal systems. > > I think the non-FS part of math contains, at least, the method of proof > by contradiction, along with other techniques that go beyond > "intuitionist" methods. There are various mathematical behaviors > mathematicians engage in that are not directed by or limited within the > confines of some particular formal system. As I've said before, it is > this ability to hop about symbolically/semantically/referentially that > makes math "more" than formal systems. > > And in many ways, this can be used to help justify the idea that math > _is_ reality, because math, like reality, doesn't seem bound within any > particular set of fixed rules. There always seems to be a way to > puncture the formalism and get at some deeper layer underneath. > There's > always a way to successfully break the rules, to > reinterpret/remanipulate the situation to one's benefit. So, while > it's > reasonable to say "reality is math", it is not reasonable to say > "reality is a formal system". > > -- > glen e. p. ropella, 971-219-3846, http://tempusdictum.com > > > ============================================================ > 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
