Glen,
> 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").
You can stay in the system. Then there's only symbols. Whoever said that
it was allowed to go outside the symbols?
And if you analyze one formal system on a higher level formal system,
then, there again, only symbols.
Everything else is philosophy (this is barebones formalism I am
advocating here - but then again - why not? you have to give reasons for
assuming more).
> 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.
You have not demonstrated the above. you have just shown that there is
undecidable stuff in some formal systems (it goes for all of course) -
but then again - whoever says that it has to bottom out somewhere? Maybe
formal systems is all there is.
> I don't believe this requires a prior philosophy of math.
It requires rejecting formalism.
> 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 ... ;-),
Ok I would agree with this - I see what you mean now. The machine can
see this with self-analysis.
I just reject the notion of some understanding "beyond the machine"
which is usually invoked, but I see that this is not what you mean.
> 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.
I agree 100%.
> 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.
Yes, very interesting, this hopping is just what would be required! But
I think in computer science just this is being done! I mean, an
operating system is doing nothing but switching contexts, right?
But only in a haphazard way (trying to optimize processing throughput) -
if you could devise a mechanism for formal-system jumping in a directed
way depending on environmental requirements, I think you would have
solved the problem of Artificial General Intelligence. It seems to be
difficult *grin*
> 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.
Here I disagree - you are reifying the word "math" - but the collection
of all formal systems is not a thing which is good to speak about I think.
"the dao that is named is not the eternal dao" ;-))
> I think the non-FS part of math contains, at least, the method of proof
> this ability to hop about symbolically/semantically/referentially that
> makes math "more" than formal systems.
We have not formalized this hopping about, but it surely is
formalizable. We humans think we can hop about as we like because we
live in tightly constrained environments (our universe, more
specifically Earth, heavily industrialized/civilized/conformed to
primate living requirments) - I would guess our cognitive systems would
crash if sufficiently alien environments were provided (probably
dropping you on Pluto in a spacesuit would be enough to make most people
go insane). So I think this "unmechanistic" jumping is an "inside view"
cognitive illusion.
> 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".
That is a deep question you are posing here (or a deep assertment you
are making *grin*); it is the crux of the matter.
So, we have the hypothesis that in the end it all boils down to formal
systems (=mechanism; which is nicely defined via computability);
or that it somehow goes beyond the formal - but what should this be? I
wonder...
Of course, computability (I equate it with mechanism here) has one thing
speaking for it: the Church-Turing Thesis. This is a deep principle,
requiring much thought. I would like to end on this philosophical note.
Thanks for your remarks Glen, very stimulating!
Cheers,
Günther
--
Günther Greindl
Department of Philosophy of Science
University of Vienna
[EMAIL PROTECTED]
Blog: http://www.complexitystudies.org/
Thesis: http://www.complexitystudies.org/proposal/
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