Re: Fw: [ROSEN] Mathematical logic, computability, and "rigor"

[Calvin Ostrum <***>]

On 8/11/05, Judith Rosen <***> wrote:
> Calvin Ostrum wrote: > Zermelo-Fraenkel set theory, actually
>  
> AL: This was an inadvertent typo.

It looks more like what someone who had heard the phrase, but
never read about it, would type.

> It can only nitpick on syntax, but cannot verify semantics.

If you mean, cannot in principle, ever, what is the proof of that?
 
> Heuristics is noncomputable. 

And the proof of that is?

> This is "naive set theory" in the sense of Paul Halmos.
> It is the standard mathematical usage.

I am aware of Paul Halmos's usage in the eponymous
book.   But how is the set theory formulated by ZFC naive
in this sense?  It is explicitly designed to avoid the 
inconsistencies that would result from a straightforward
attempt to include "naive" axioms of comprehension.

> > Calvin Ostrum wrote: What useful mathematics cannot be done "purely
> syntactically".
>  
> AL: We cannot syntactically come up with intuition.  In other
> words, "how to come up with the Eureka! to prove a theorem" is
> impredicative. 

I really have no idea now what "impredicative" could possible mean
in any kind of rigorous sense!  It seems to mean "magical, mystical,
wonderful, non-reductionist".  But the fact that we cannot just churn out  
all the theorems we want does not mean that mathematics cannot 
be captured in ZFC.    That is our failing, not the failing of ZFC. 

> Once the idea is formed, the proof itself is of
> course done syntactically and formally.

So in other words, virtually all useful mathematics *can* be captured in ZFC 
after  all, apparently.    Okay, I'm glad we have that settled.

> That was slightly flippant on my part.  Of course a lot
> of good science came out of pure syntax

What science "comes out of", and the mathematical systems that
are used in his process, are different things.   The mathematics
used without exception in serious science seems to be quite
readily captured in ZFC or other formal systems.

Also, I have not seen an answer to the question of what the space is
where this non-capturable mathematics lies, and the measure on the space,
that allows one to make any sense out of the analogy with the reals/rationals.

>  But Rosen's point was
> that so much more could be done if we go beyond the machine
> metaphor.  "Molecular biology" was his favorite oxymoron.  Once
> we go "molecular" it is no longer "biology" -- "molecular
> biophysics" or "molecular biochemistry" perhaps, and the subject
> yields useful data, but it is not biology.  "It's dead, Jim."

So what is this "so much more", precisely?