Thu, 06 Jan 2000 02:59:47 -0800
It's about time this list gets a good "oming" so here it is:
In this message I will meditate on the profound lameness of godel numbering. I
mean all it does is map a set of discrete symbols to an arbitrary set of
numbers. That, to me is lame. It, in itself, doesn't reveal anything about the
problem, It's just a different encoding. I will grant to you that encoding
things differently can, and in this case does, allow you to talk about them in
a more familiar context. Contrary to Hofstadter's claim, it destroys a certain
type of information and that is when you look at a formal system, any formal
system, you can say one of two things: The symbols in this system are mapped
to these meanings. Or you can say Aha, this is the TNT system or one of the
other systems in Hofstadter's undecidably good or horrible book which is
associated with this set of rules. When someone hands you a number you can't
at all tell what system it is a mapping of! In that sense the number, while
retaining other properties, has lost its function. It ceases to be a mechanism
and becomes an inert number. =((( I think it is possible to recover the order
of symbols within a Godel number by doing some tricks with factorization.
Hofstadter's claim about the above is that the meaning comes from an
"isomorphism".... Well OKAY... We have gone from a very strictly defined
formal system to a system that relys on "Isomorphisms". I have still, untill I
figure this out some more, lost the ability to specify exactly what I really
mean to be isomorphic to. Here's a rather snyde example: Lets say I decide to
express this issue to Hofstadter. I first set about defining a formal system
that is intended to be mappable to a human language. Lojban may pass for this.
So I then write a program that takes all the symbols of Lojban and maps them
to arbitrary but unique numbers. I then write my letter to Hofstadter,
translate it using my program into its godel number, and then hit send. Will
he ever get the message?
Well lets see... How would you attempt to isolate or catalogue an
isomorphism? Supposing the factorizing trick works and you can get a string of
symbols out of the single godawfulhuge number then what? The first thing
almost anyone would try would be the direct approach. That means you would
take the string and then count all other strings that have that same pattern
of symbol recurrance and are of the same length. When you find a match, and I
suspect you will find MANY, you add it to a list of "hits" which you will have
to manually read down and take a guess at what was meant. =P There are other
more circuitous routes that attempt to guess the rules of the symbols in the
input stream but still I don't think Mr. Hofstadter will be making heads or
tails of his mail real soon. =P
Lets look at tihs problem now, one more time before I colapse on my keyboard,
from the TUNES perspective. Lets suppose you had a large number of godel
numbers of known correct lambda calculus expressions. (Lisp is partially based
on this language). Would it be possible to write a program that takes as input
a specification of a program and then breaks it down into known routines, then
multiplies the godel numbers of the library routines togeather and finally
uses the known lambda calculus mapping to extract an executable that can be
sent streight to the processor to be executed? Could this system then be
extended to include the rules of lambda calculus itself to then be "reflexive"
as the current activity seems to be focused?
I think I have argued fairly well against the second question in this text.
The godel numbering is actually a sideshow. It does not have any use but to
express a specific theorem or data to be processed within the system. The true
meaning of these godel numbers are translated into number theory by rigorous
methods that are outside of the godel numbers. The only way to access the
meaning of the godel numbers is through these translated rules. Therefore
making a random statement such as "this string is unproovable in TNT" fails to
have any meaning because the critical piece of information is lost. IMO at
Anyway, I hope I won't have to take back most of this when I'm rested....
Drugs are good.
They automatically screen the deltas and gammas from the alphas and betas.