Le 23-janv.-07, à 15:59, 1Z a écrit :

>
>
> Bruno Marchal wrote:
>
>> Also, nobody has proved the existence of a primitive physical 
>> universe.
>
> Or of a Platonia




Call it Platonia, God, Universe, or Glass-of-Beer, we don' t care. But 
we have to bet on a "reality", if we want some progress.

Now, here is what I do. For each lobian machine I extract a "theology" 
from her personal discourses. The discourses of the machine refers to 
something NOT describable in its language (actually "truth"). That is, 
each correct lobian machine refers to a reality which has no name for 
her.
Now, it happens that a very-rich machine, like ZF, *can* refer and give 
name to the unnameable reality of a simpler correct lobian machine. So 
the machine ZF can name and prove the whole theology of a simpler than 
herself correct lobian machine. For example ZF can define "arithmetical 
truth", which is just the "Platonia/God/Universe..." of the (rather 
simple) lobian machine PA. If you add to the machine ZF (seen as any 
reasonable theorem prover of the ZF theory) a very simple (trivial) 
inductive inference ability, she can bet/hope that she is, not only 
lobian, but "correct", and so she can lift (but NOT PROVE, 'course) 
PA's theology to herself, and "knows" (relatively to her hopes!) that 
her "big reality" has no name.

Note this: everyone (human) know that PA is correct. Everyone (human) 
can name PA's Platonia. This is enough to prove that as a lobian 
machine every human is more rich than PA. Now, I don't know if I am 
richer than ZF. Not only ZF cannot name "set-theoretical truth", but I 
am not sure human can do that. A case can be given that ZF is already 
too much rich. Set theoretical truth, unlike arithmetical truth *is* a 
bit problematic.


Note this: all the theologies of all consistent lobian machines and 
even of all consistent lobian entities (like "angels", those 
generalized NON-machine provability system like Analysis+omega-rule) 
are isomorphic. They are all described by G and G* and the intensional 
variants: the 8 hypostases (with Plotinus' vocabulary). But the modal 
connector "B" is an indexical: it is a notion of third-person "I". It 
means ZF when B is the provability in ZF, and it means PA when B 
represents the provability in PA (like "I" = Bruno when asserted by 
Bruno, and John when asserted by John). But all third-person "I" obeys 
the same hypostase-logics,  where  "I" refers to any correct lobian 
entity (machine or not).


Remark: I say that PA is simpler than ZF. By this I mean that 1) you 
can translate any theorems of PA in ZF, and 2) ZF can prove those 
theorems. Put in another way, it means that ZF contains PA, modulo that 
translation.
Now ZF is not simpler that PA: this means the reverse is not true: 
there are theorems of ZF that either you cannot translate in PA's 
language, and there are proposition of ZF that you can translate in PA 
but that PA cannot prove. Example: PA cannot name its "platonia", but 
ZF can name PA's platonia. PA can name its own consistency, but cannot 
prove it. ZF can name PA's consistency and prove it (but 'course, 
cannot prove it).

Last and absolutely important remark: I have just said that ZF can 
prove the consistency of PA. And PA cannot prove the consistency of PA, 
making ZF more powerful than PA. The point is that PA can prove that! 
That is, PA can prove that ZF can prove the consistency of PA.  But PA 
has no reason at all to trust or even just "understand" ZF.
This means that PA can simulate ZF, like the non-chinese in Searle's 
room can "talk" chinese, actually without any understanding. Like I can 
solve Einstein's Gravity Equation, if you give me a correct description 
of its brain and the time to process it (!).
So the distinction between computability/emulabity and PROVABILITY is 
already enough for preventing us to do "Searle's fundamental error": 
its confusion between genuine personal understanding of chinese by the 
emulated chinese, and the non understanding of the simulator itself.
Searles' error is a fundamental error to meditate on. Usually I don't 
insist because I tend to consider that Hofstadter and Dennett, in 
Minds'I, are quite good and sufficient on it.
(Please, note that when I say a philosopher is wrong, this should be 
taken as a compliment; and sometimes the error is fundamental, I will 
probably refer a lot of times to that "Searles' Error"). Science is 
just philosophy made refutable.

As computer/simulator, both PA and ZF are universal and equivalent. As 
believer or theorem prover, ZF is far more powerful (although 
incomplete and necessarily so by Godel II) than PA. The price of 
universality in computation/simulation (Church Thesis) is the lack of 
universality in theorem proving, belief systems, etc. cf the Fi and Wi: 
I'l come back on this.

Note that PA is described here as "simple", but actually PA is rather 
gifted, and I could argue that 98% of today math, including 98% of 
Ramanujan's work, belongs to its discourse. It is possible to build or 
define lobian machine much simpler than PA, but PA is more easy to 
describe, and so I take her as a simple example of simple machine, but 
this should be relativize a little bit. See any texbook in mathematical 
logic for a description of PA, or click here:   
http://www.ltn.lv/~podnieks/gt3.html

Bruno


http://iridia.ulb.ac.be/~marchal/


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