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From: Marchal <marchal.domain.name.hidden>

Date: Fri Jul 9 06:51:03 1999

Romanos Avaqian wrote:

*>Before long talks about consciousness, may be there is a sense to discuss at
*

*>first who/what is the subject of perception ? The world is the object, the
*

*>intellect is the tool, but who/what is the subject? Who perceives ?
*

*>The second important question: can the subject be perceived ? Otherwise
*

*>saying: can the subject become an object for himself ?
*

In Theoretical Computer Science,

there is a traditional way to handle self-reference.

Basicaly self-reference is builded upon the so called

second recursion theorem by Kleene 1938 see (1).

This theorem says that for any computable

transformation, you can build

a program which applies that transformation to itself.

The theorem is constructive. Given the

the code of the transformation T you can build the

program which applies T to itself. For exemple, in case T

computes the identity function, you get a self-reproducing program.

The second recursion theorem is also true about the set of

metaprograms and the metametaprograms ... This has been

shown by John Case, a modern 'specialist' on self-reference: see his

WEB pages for more links to self-reference:

http://www.cis.udel.edu/~case/index.html

This makes for exemple possible the building of self-referential

collection

of machines, like self-regenerating nets of machines, similar

to hydra or planaria.

In that sense the machine can be an object to itself.

Suppose a machine has some ability to prove, like an automatic

proving machine or an automated deduction system.

What can an automated deduction system proves about itself ?

What can a consistent automated deduction system says about itself ?

An incredible use of, still the second recursion theorem (or his

formalised version known as the diagonalisation lemma) has make

it possible for Solovay to settle that question, in 1985, after

Godel 1931 and Lob 1955 deep breaktroughs. At least he settles

that question concerning sound machines (or theories) which believe

(or proves theorems) in

classical logic and elementary arithmetic.

The results are astonishing : not only there is a decidable

axiomatic system (G) which formalises completely what the machines

can prove, but there is also an axiomatic system (G*) which formalizes

completely what is true about the machine (including what the

machine is unable to prove). (all this at the propositional level!).

NOW THE BIG PROBLEM: we have hardly talk on

any suject here. The classical

theory of self-reference talks only about what a

machine-object can correctly tell

about itself seen as an object. It is a sort of third-person

self-reference.

Only intellect and object are at work until here.

So, how to define an subject coherently with the classical

self-reference theory ?

Believe it or not, Romanos, but an answer

can be found in Plato's Thaetetus.

Or at least an attempt toward an answer.

The basic idea is to define the subject

as the one who knows (as opposed to the intellect

who proves or communicates convincingly), and then

to define "knowing p" by "justifying (proving) p AND p".

Another basic motivation for this strategy can perhaps be provided by

some interpretation of

Descartes' first meditations or by the traditional idealist use of

the dream (the old natural virtual reality!), in Pythagore's

and Platon (still the Thaetetus) philosophy

or in some Indian idealist philosophical school (think

about the yogavasistha(2).

... or Lewis Carroll.

I am aware that such a definition of knowledge is still very

debated among philosophers.

Nevertheless, in that case, with such definition,

it is an exercice, even an easy

one for those who read at least that wonderfull recreative book

"Forever Undecided" by Raymond Smullyan (another professional of

classical self-reference), to show that

the subject cannot be a object for itself.

This, I hope provide an answer to your question, and a path

(the godelian one) toward some communicable light bearing on

similar question.

More details and references are given in http://iridia.ulb.ac.be/~marchal.

Note also that it is not necessary to believe in Thaetetus' strategy

to use it to invalide arguments, as I do for Lucas-Penrose

type of argument in http://iridia.ulb.ac.be/~marchal.

Even for non-computationalist such an approach can provide an

*etalon philosophy* useful for possible comparisons between

various philosophies.

What about "the observer" ? I give reasons to *define* it

by *I observe p* = "I justify p AND p is consistent", for

a class of atomic "verifiable" proposition p (in computer

science those are the sigma-1 propositions).

This is a way to internalise the Many-Consistent-Worlds

idea, including the fact that we are searching a probability

defined on the set of those worlds (or states).

In that case we find indeed a sort of probability

logic rather similar to classical Birckhof-Von Neumann Quantum

Logic.

This gives tools for extracting a precise measure on the

*set of relative

computational histories* having a welcome Quantum smell.

Thanks to the difference between G and G*, we get two Quantum

Logics, one for the observable and communicable propositions,

i.e. *the physical propositions*, and one for the observable

but non communicable propositions. This last logic is a good

candidate for a Logic of Qualia.

Bruno

---------------------------------

(1) Kleene, S. K., Introduction to metamathematics, 1952, North-Holland,

page 352.

(2) A nice (occidental) relevant book here is Wendy Doniger O'Flaherty's

"Dreams, Illusion and other Realities". The University of Chicago Press

1984.

Other reference in http://iridia.ulb.ac.be/~marchal.

See also my 1992 paper "Amoeba, Planaria, and Dreaming Machines".

Received on Fri Jul 09 1999 - 06:51:03 PDT

Date: Fri Jul 9 06:51:03 1999

Romanos Avaqian wrote:

In Theoretical Computer Science,

there is a traditional way to handle self-reference.

Basicaly self-reference is builded upon the so called

second recursion theorem by Kleene 1938 see (1).

This theorem says that for any computable

transformation, you can build

a program which applies that transformation to itself.

The theorem is constructive. Given the

the code of the transformation T you can build the

program which applies T to itself. For exemple, in case T

computes the identity function, you get a self-reproducing program.

The second recursion theorem is also true about the set of

metaprograms and the metametaprograms ... This has been

shown by John Case, a modern 'specialist' on self-reference: see his

WEB pages for more links to self-reference:

http://www.cis.udel.edu/~case/index.html

This makes for exemple possible the building of self-referential

collection

of machines, like self-regenerating nets of machines, similar

to hydra or planaria.

In that sense the machine can be an object to itself.

Suppose a machine has some ability to prove, like an automatic

proving machine or an automated deduction system.

What can an automated deduction system proves about itself ?

What can a consistent automated deduction system says about itself ?

An incredible use of, still the second recursion theorem (or his

formalised version known as the diagonalisation lemma) has make

it possible for Solovay to settle that question, in 1985, after

Godel 1931 and Lob 1955 deep breaktroughs. At least he settles

that question concerning sound machines (or theories) which believe

(or proves theorems) in

classical logic and elementary arithmetic.

The results are astonishing : not only there is a decidable

axiomatic system (G) which formalises completely what the machines

can prove, but there is also an axiomatic system (G*) which formalizes

completely what is true about the machine (including what the

machine is unable to prove). (all this at the propositional level!).

NOW THE BIG PROBLEM: we have hardly talk on

any suject here. The classical

theory of self-reference talks only about what a

machine-object can correctly tell

about itself seen as an object. It is a sort of third-person

self-reference.

Only intellect and object are at work until here.

So, how to define an subject coherently with the classical

self-reference theory ?

Believe it or not, Romanos, but an answer

can be found in Plato's Thaetetus.

Or at least an attempt toward an answer.

The basic idea is to define the subject

as the one who knows (as opposed to the intellect

who proves or communicates convincingly), and then

to define "knowing p" by "justifying (proving) p AND p".

Another basic motivation for this strategy can perhaps be provided by

some interpretation of

Descartes' first meditations or by the traditional idealist use of

the dream (the old natural virtual reality!), in Pythagore's

and Platon (still the Thaetetus) philosophy

or in some Indian idealist philosophical school (think

about the yogavasistha(2).

... or Lewis Carroll.

I am aware that such a definition of knowledge is still very

debated among philosophers.

Nevertheless, in that case, with such definition,

it is an exercice, even an easy

one for those who read at least that wonderfull recreative book

"Forever Undecided" by Raymond Smullyan (another professional of

classical self-reference), to show that

the subject cannot be a object for itself.

This, I hope provide an answer to your question, and a path

(the godelian one) toward some communicable light bearing on

similar question.

More details and references are given in http://iridia.ulb.ac.be/~marchal.

Note also that it is not necessary to believe in Thaetetus' strategy

to use it to invalide arguments, as I do for Lucas-Penrose

type of argument in http://iridia.ulb.ac.be/~marchal.

Even for non-computationalist such an approach can provide an

*etalon philosophy* useful for possible comparisons between

various philosophies.

What about "the observer" ? I give reasons to *define* it

by *I observe p* = "I justify p AND p is consistent", for

a class of atomic "verifiable" proposition p (in computer

science those are the sigma-1 propositions).

This is a way to internalise the Many-Consistent-Worlds

idea, including the fact that we are searching a probability

defined on the set of those worlds (or states).

In that case we find indeed a sort of probability

logic rather similar to classical Birckhof-Von Neumann Quantum

Logic.

This gives tools for extracting a precise measure on the

*set of relative

computational histories* having a welcome Quantum smell.

Thanks to the difference between G and G*, we get two Quantum

Logics, one for the observable and communicable propositions,

i.e. *the physical propositions*, and one for the observable

but non communicable propositions. This last logic is a good

candidate for a Logic of Qualia.

Bruno

---------------------------------

(1) Kleene, S. K., Introduction to metamathematics, 1952, North-Holland,

page 352.

(2) A nice (occidental) relevant book here is Wendy Doniger O'Flaherty's

"Dreams, Illusion and other Realities". The University of Chicago Press

1984.

Other reference in http://iridia.ulb.ac.be/~marchal.

See also my 1992 paper "Amoeba, Planaria, and Dreaming Machines".

Received on Fri Jul 09 1999 - 06:51:03 PDT

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