On 02 Sep 2010, at 17:02, David Nyman wrote:

On 1 September 2010 21:51, Bruno Marchal <marc...@ulb.ac.be> wrote:

In
other words, it would appear that he cannot, with pure logic, rule out
the possibility - even to himself - of Logical-David being merely a
zombie.

No. Even the zombie can see that he cannot. That's why the
self-referentially correct machine cannot be a zombie.

Yes, my emphasis here was indeed **with pure logic**.  That is to say,
the "purely logical part" - i.e. the believer - begins by suspecting
that it perhaps can't rule out the possibility that it is *only* that:
i.e. pure logic.

Not at all. He has a relation with truth. He begins by confusing its belief with truth, but introspecting itself, it discovers the gap between truth and its beliefs. So that he discovers the gap between beliefs (the "logical system") and knowledge (by definition: those beliefs which are, some perhaps serendipitously, others perhaps necessarily, true.





But the subtlety is that it can also realise that
this very possibility begs the question of its own self-referential
integrity.

Yes. Indeed.


 That is, it would place into doubt its very "existence for
itself" as something (i.e. anything whatsoever) metaphysically
distinct from the substrate of logic considered to be constituting it.

Good.




The following dilemma is then presented: either a) there simply *is
no believer* in any way metaphysically distinct from the substrate in
general,

Certainly not.


or b) the truth of such distinctness is for the believer
alone to assert.

Exact.




But then, since the unpalatable consequence of
option a) would be the annihilation of both the Cat and the Grin, the
believer (thankfully) realises that "self-referential correctness"
mandates the truth of its own existence-for-itself.  What a relief!

I think so.




Why can this be so hard to see?  This thought was what motivated my
original post - i.e. that the reason (or at least a major reason) why
all this can seem so elusive is that we continually seek to escape
self-reference by adopting - per impossibile - some maximally analytic
"objectivised" viewpoint: i.e. the reductionist-god's eye view, or
"observer at infinity".  Although success in achieving such a "view"
should entail the consequence of entirely banishing any
self-referential distinctness we possess, somehow we can fail to
appreciate this, and the Cheshire Cat-like apparition of the "objects
of experience" remains in full view. How is this?  Because we forget
that any "view" purporting to be both entirely analytic and *outside
of everything* must, by that very restriction, have abandoned any
legitimate grasp of *internally-referenced* categories predicated on
integration, composition, or synthesis.

That sentence is long ;-)



We just go on "projecting"
all this from afar, just as though we had not absented ourselves from
the scene.  Such metaphysical absent-mindedness tricks us into begging
the question of just how such synthetic categories could in fact
acquire any transcendence over a putative analytic substrate.  In
effect, this kind of metaphysical circularity is "hard-wired" into our
naturalistic modes of thought - though we may blind ourselves to
self-reference, we cannot escape it.

I think so. As far as I don't misunderstand. Give me time to digest.




You have to study Gödel and Tarski theorem, or a result by Kaplan and
Montague, recasted in the Solovay logics. Smorynski wrote a paper "50 years of arithmetical self-reference", a rather long time ago. Boolos 1979 and 1993 consecrated a chapter to the "knower" (S4Grz). A student and friend of mine has formalized Bp & Dp (the Z, Z1, Z* and Z1*) logics, but the case for S4GRz1, and the X, X1, X*, X1* logics remains unsolved. There are theorem
prover for those logics, and so by an indirect argument we know them
formalizable, but no one has found the axioms yet. Note that all those logic are non effectively soluble, once extended at the modal predicate level. The hypostases give a knowledge of the believer, the conscious knower, the observer, the "feeler" each with they communicable and non communicable part, from which you can derive, the observable, the non observable, the
feelable and the non feelable, well many things, including quanta and
qualia.
Weakness: hard mathematics.
Ask any question if you feel so. I am aware it is hard and ultra- subtle stuff. Well, that is why only technics can handle those self- references.

It is, as you say, remarkable that there exist detailed systems of
self-referential logic that can capture such subtleties, and make
rigorous the distinction between formal and non-formal parts.  I am
grateful for your continued perseverance and patience in affording me
even the most basic insight into them, and I only wish I had the sheer
tenacity to get to grips with them in the extended technical detail
they demand, as you do.  But you, after all, are a logician and I am a
mere quibbler.  Nevertheless, it intrigues me that my quibbling
occasionally seems to lead me somewhere in the vicinity of these
notions, so I won't abandon it entirely!


I appreciate very much your patience. Got the feeling you see the thing.

Let me try to take the opportunity to try to give you the "arithmetical" thing, and why logic+arithmetic force the self- referential logic to distinguish between the formal (body, syntax, finite machine, program, word, number, the 3-thing) and the informal (truth, person, knowledge, the 1-thing).

Allow me to be not exact on the details, for example Gödel has chosen PRINCIPIA MATHEMATICA as pet formal "rich" theory, and I choose Peano Arithmetic (like all logicians in all textbook). Gödel and Tarski have shown remarkable facts:

There are true arithmetical sentences that PA cannot prove.
PA can talk about its own probability.
In particular PA can prove that IF PA will never prove a falsity, that is, IF PA is consistent, then PA cannot prove that PA is consistent.
That remains true for all consistent extension of PA.
That PA can prove all this. In particular PA proves that "IF PA is consistent then PA cannot prove that PA is consistent".
PA can talk about provability, but cannot even name truth.
That PA can, nevertheless, name a transfinite approximation of truth.
PA can prove that "as far" as PA is "correct" then G* apply to it. So, as far as I am true, beliefs coincide with knowledge. Some effective extension of PA will be able to "see" that PA's knowledge is not definable by PA (and so is informal), and obeys a different logic (an S4 logic instead of G).

I will think about trying to be even more clear and pedagogical. I may sum up that I do agree with your analysis, by my own intuition/ introspection (that I try to convey with mechanism, and uda, ...), but that PA seems to agree too, entailing that all effective (machine) extensions of PA would agree to :)

Truth is Plato's god. It is the undefinable object of research of the fundamental inquirer. Truth about numbers is far beyond numbers. But numbers can already see that.

Bruno










On 01 Sep 2010, at 20:03, David Nyman wrote:

On 1 September 2010 09:21, Bruno Marchal <marc...@ulb.ac.be> wrote:

How does my experience of

dreaming of a tree connect to numbers?  What is it that generates my

experience of a tree from the brutely existing substrate of numbers?

Well, from the true but non communicable part given by the self- reference

logic of self-introspecting (ideally correct) programs (machine, numbers,

theories ... words are used in a large sense here).

Rex's question excerpted above, and Bruno's response to it, seem to
relate directly to the topic I had in mind in my original post.
Speaking, as it were, somewhat in Bruno's rather "large" sense, the
"brutely existing substrate of numbers" might correspond to that
particular perspective on the Real which is characterised by
abstraction to what I called the pole of maximal fragmentation - i.e.
the role presumably occupied by the quantum field and its
manifestations in current physical theory.  "The self-reference logic
of self-introspecting (ideally correct) programs (machine, numbers,
theories ...", would then be seen in the role of a combinatorial logic
operating over this domain - that occupied, in physical theory, by
whatever are supposed to be considered the fundamental relations
between "physical" ultimates.

The key subtlety then devolves on "the true but non communicable
part".  This seems very difficult to state concisely.  Perhaps what
makes it elusive is that first one has to appreciate that the
self-introspecting logic is (somehow) already capable of grasping that
certain of its beliefs have the characteristic of implying to the
believer, in a certain sense, the "metaphysical distinctness" of their
referents.  To see this, let us consider "Logical-David" - i.e. that
aspect of David that is purely a manifestation of "self-introspecting
(ideally correct) programs".  This Logical-David possesses - i.e. is
(partially) constituted by - certain self-referential beliefs that
mediate relations between "himself" and the "objects of his
experience".  He is already, within the constraints of this purely
logical substrate, capable of demonstrating consistent commitment to
such beliefs, and to their putative referents; indeed this is what
motivates any communicable judgement or statement whatsoever that he
is capable of producing about them. There is something more, however.
There is something that already seems to him to transcend this purely
logical "substrate", something somehow metaphysically distinct, that
seems to arise from its peculiarly self-referencing character.  It
seems somehow to be those very "objects of experience" themselves.

On reflection, however, Logical-David is (just) able to see that all
these considerations can still be confined within the constitutive
substrate of numbers and combinatorial logic.  Or to put it another
way, as someone notoriously said, "he would say that, wouldn't he"?
Whatever this extra something might, or might not, be, he is incapable
of communicating it directly.

Yes. It is the truth as such that he cannot communicate. The third person description can be confined in the combinatorials, but not the truth. Like
the truth of feeling to be the one reconstituted at here or there in
self-multipliation. A part of truth is livable, but non communicable as
such.

Consequently, purely "logically", he
must (just) concede that any such putative metaphysical distinctness
could still be, from a purely "logical" standpoint, chimerical.

No. G* extended properly G. There is two self-referential logics. The true
one, obeying G*, and the communicable part G.
The machine can expect that the same thing (truth) obeys different logic (first person, third person). Incompleteness protects the machine from
confusing the views. If he takes G* minus G as chimerical, he becomes
inconsistent, or it lacks self correctness.

 In
other words, it would appear that he cannot, with pure logic, rule out
the possibility - even to himself - of Logical-David being merely a
zombie.

No. Even the zombie can see that he cannot. That's why the
self-referentially correct machine cannot be a zombie.

And indeed, Logical-David is, precisely, such a zombie.

In a sense you are right. But such a third person describable "logical david", is more akin to David's body, than David the (first) person, which (by definition) is connected to the truth. You are confusing Bp, and Bp & p.
The third person self and the first person self.

Actually, in this somewhat etiolated form he really should be
considered more an intriguing interpretation of the
mathematico-logical substrate than a person, as it were, in his own
right

Can there be no escape from this seemingly doleful logic?  Yes, if
there is *in fact* a David whose personal individuation is able to
transcend its merely logical embodiment.

That's what G* and its intensional variants offer on a plate.

And such transcendence
indeed implies a metaphysically distinct, direct grasp of certain
truths beyond their mere logical implication; some sort of personal
integration or synthesis - apotheosis, even - correlated with, but
irreducible to, any substrate considered in its purely
analytical-combinatorial aspect.

Exactly. You put things very well. But the G-G* separation makes S4Grz1, the logic of the soul (Bp & p) a logical of a non formalisable person. You cannot translate in the combinatorial or in term of numbers the "p" of Bp &
p. The soul has no name, no 3-descriptions, from its point of view.
Yes, S4Grz, S4Grz1, X1, and X1* are quite remarkable, those logics formalize the discourse of beings which are themselves NO formalizable, nor even nameable by the machine, except by some vindication like "could you please
take care where you put your foot".


In short, for such transcendent
individuation to be the case, there must *actually exist* a
first-person David who is conscious,

David the soul. Again, you get it for the machine by the Theatetical idea of defining the soul by the knower, and the knower by the connection between the believer (a mechanical body) and God, or simply Truth. It is the passage from G1 to S4Grz1. (the "1" has to be added in the for the machine in the comp frame). It *is* the passage from Bp to Bp & p. It is "p" (the truth of
p) which is responsible for the non formalizability.

as well as merely logically
possessed, of the objects of his experience.

That's David-the-body, or the dynamical-believer machine. That's the one
described by G and G*.

And as to the truth of this - of course - only he would know.

As I said. You put *very well* the thing. You add the "truth" at the right place, at an infinitesimal nuance I don't want to bother you with now.


David (both of him)

You will end up 8 of you. Or 4 + 4.infinity, Oh! well, thare are many ays to counts them. I say 8 for the 8 main intensional variant of "third person
belief":
p
Bp
Bp & p
Bp & Dp
Bp & Dp & p
I have underlined those variants who corresponds to non formalizable beings. I have put in italic those who split in two, due to the G-G* gap. So you can count: there are 8. Self-referential machine have to live this that. They have already 8 quite different views on the truth. [or 4 + 4.infinity, because you have all the variant with B^n p & D^m p in place of Bp & Dp, and
where B^5 p is BBBBBp ... ].
You have to study Gödel and Tarski theorem, or a result by Kaplan and
Montague, recasted in the Solovay logics. Smorynski wrote a paper "50 years of arithmetical self-reference", a rather long time ago. Boolos 1979 and 1993 consecrated a chapter to the "knower" (S4Grz). A student and friend of mine has formalized Bp & Dp (the Z, Z1, Z* and Z1*) logics, but the case for S4GRz1, and the X, X1, X*, X1* logics remains unsolved. There are theorem
prover for those logics, and so by an indirect argument we know them
formalizable, but no one has found the axioms yet. Note that all those logic are non effectively soluble, once extended at the modal predicate level. The hypostases give a knowledge of the believer, the conscious knower, the observer, the "feeler" each with they communicable and non communicable part, from which you can derive, the observable, the non observable, the
feelable and the non feelable, well many things, including quanta and
qualia.
Weakness: hard mathematics.
Ask any question if you feel so. I am aware it is hard and ultra- subtle stuff. Well, that is why only technics can handle those self- references.
Bruno Marchal







On 31 Aug 2010, at 19:36, Rex Allen wrote:

On Mon, Aug 30, 2010 at 11:11 AM, Bruno Marchal <marc...@ulb.ac.be> wrote:

So "idealist accidentalism"...the view that what exists is mental, and

that there is no underlying process that explains or governs this

existence.

If idealist accidentalism is correct then there is no theory at all.

Well, I'd have to hear your definition of "theory" and what the

conditions are for its existence.

The existence of a theory is usually not the object of the theory, but of a

metatheory.

In some case the metatheory can itself be an object of the theory. For

example zoologists are animal (but botanist are not plant). Since Gödel we

know that the theory "Peano Arithmetic" can be studied "in" Peano

arithmetic. And monist philosophies makes mandatory that the theory and/or

the theoretican has to belong to the collection of objects or phenomena of

the theory. Physicists do obey to the laws of gravitation for example. A

physicist of masse m will attract a physicist of mass M with a force

proportional to mM/(square of the distance between two physicists). of

course that force is negligible compared to the natural repulsion that a

physicist can or cannot have for a colleague ...




So obviously something exists...my conscious experience of this

moment.

That is obvious for you. I have to postulate it.

Unless you postulate we are the same person?

I can agree with that, at some level, but you waould not refer to "this

moment". I am not sure what you mean by "moment" with idealist accidentalism

(IA).



This experience is a multifaceted thing...in that there are

many "things" I am conscious of in this moment.

But this is true of dreams as well.  I am conscious of many things in

a dream, but those aren't things that exist outside or independently

of the dream.

In which theory. Such a sentence seems to assume a lot, if only to make some

sense. If IA is correct, words like "world", "outside" refer to what?



So what accounts for the dream?  Numbers?

In the theory "digital mechanism", aka "computationnalism", we can argue for

this, indeed.



How does my experience of

dreaming of a tree connect to numbers?  What is it that generates my

experience of a tree from the brutely existing substrate of numbers?

Well, from the true but non communicable part given by the self- reference

logic of self-introspecting (ideally correct) programs (machine, numbers,

theories ... words are used in a large sense here).



Why should numbers give rise to my dream experience of a tree?

Obviously I can use numbers to represent the tree...in the sense that

I can use saved numerical measurements to "re-present" the tree to my

self...if I can remember how to interpret the measurements.  And I'm

even willing to grant that I can use numbers to represent my

experience of the tree.  But representation is just the re-presenting

of something to your conscious experience, which is not at all the

same as explaining the fact of that experience.

The fact of experience is given by the true fixed point of the

representation, like a map of the USA, when situated in the USA will have a

representing point superposed on the real point.





But idealist accidentalism is a theory (even if vague)

So there is no theory, and there is one theory.

So 0 = 1.

Contradiction.

So idealist accidentalism is refuted.

I think you should have your logician license revoked...

I will not insist on that littel reasoning. Was just trying to shortly

points that IA makes little sense for me.




You may save it by insisting that idealist accidentalism is not a theory.

It

would be a mere philosophical injunction of the type "dont' ask, don't

search".

I think it is a just a recognition that Agrippa's trilemma and the

principle of sufficient reason lead to infinite levels of infinite

regress.  Which I take as a sign that there's something wrong with

that type of interpretation of our conscious experience.



When put in computer science terms (which computationalism invites naturally

to do), we inherit of the fixed point solutions of recursive equation.

I have no problem with people trying different kind of theory, but to posit

consciousness at the start (or matter, actually) does not satisfy me. As I

said it prevents further research. I understand that feeling (consciousness

cannot be explained), but I can at least explain why machine/ numbers develop

discourse invoking similar failure feeling about their own

consciousness/consistency, or true but non provable predicate on themselves.

Bruno


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



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