Re: Mathematical logic, computability, and "rigor"

[Calvin Ostrum <***>]

On 8/11/05, Judith Rosen <***> wrote:

>  The conclusion that
> begins to suggest itself to me, in the face of these facts, is that neither
> of you are really interested in understanding the work.

A completely unfounded conclusion in my case.  Although it is not
Rosen's work, first and foremost, I am interested in.   But I think it is
clear that all life (defined as it currently is, vaguely, in terms of a 
thing which in some sense causes itself, along the lines of for example
Varela's "autopoiesis") could be nothing more than entirely 
computable mechanisms (even if the life we actually know is not), so 
if  someone claims otherwise, it is important to see precisely 
what their argument is. 

 So, perhaps you can
> help me with something that I have already asked the list subscribership to
> see if they could attempt: Let's see if you can come up with any real way to
> "debunk" the scientific conclusions my father arrived at. 

Before any conclusion can be debunked, it must be made precise what
it actually is.   There cannot be vague handwaving about things like 
"the reductionist mode" and the like in the conclusions that we are
examining.  Note that Goedel's Theorem, by contrast, has been given
a very rigorous formulation many times, including by Goedel himself.
And it has been reworked many times and rendered in a very perspicuous
format, without any loud bragging about its deeply revolutionary nature.
   
> What I'm after is a (relatively) quick way to solve, once and for all, the
> arguments over whether or not these theories are indeed representing aspects
> of the universe that are not included in current scientific capability. I
> trust you both will agree that if Robert Rosen is right about that, it
> behooves humanity to know it-- and begin work developing new scientific
> capability that will rectify the situation? And, if it is possible to prove
> that he was not correct in his conclusions, then let's do so and save
> ourselves a whole lot of breath. 
>   
> What do you say? Are you game? 

Of course.   So I think we should start by developing a rigorous
account of precisely what is meant by each term in the statement
I gave (or perhaps a corrected or related statement).  This will render 
the statement into a form that can either be proven or disproven.  

But this is going to take a fair bit of work.   And I for one cannot put 
all of my eggs in one basket, especially a sort of basket that is
essentially ignored by egg carriers everywhere.  For example,
when talking about things such as the infinitesimal amount of
mathematics that can be "captured" in formal systems, I cannot
afford to be ignorant of the nature of PA and ZFC with respect to
what they can "capture".  So the work I myself manage to do on
this is going to proceed slowly.