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
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