At 15:33 03/06/04 -0700, George Levy wrote:
I reread your post of 5/11/2004 and it raised some questions and a possible paradox involving the idea that the "notion of first person is absolutely not formalizable." (see below, for a quotation from your post)
<< It may be that using the observer as starting points will force White Rabbits to be filtered out of the
<< observable world
>>And again I totally agree. It *is* what is proved in my thesis. I have done two things:
>>1) I have given a proof that if we are machine then physics must be redefined as a
>>science which isolates and exploits a (first person plural) measure on the set of all
>>computational histories. The proof is rigorous, I would say definitive (unless some systematic
>>error of course), although provably unformalizable (so that only 1 person can grasp it).
>>2) I provide a mathematical confirmation of comp by showing that (thanks to Godel,
>>Lob, Solovay ...) we can literally interview a universal machine, acting like a scientist
>>---by which I mean we will have only a third person discourse with her. BUT we can
>>interview her about the possible 1-person discourse. That is a "tour de force" in the sense
>>that the notion of first person is absolutely not formalizable (and so we cannot
>>define it in any third person way). But by using in a special way ideas
>>from Plato's Theaetetus + Aristotle-Kripke modal logic + Godel's incompleteness
>>discovery make the "tour de force" easily tractable.
>>Here I can only be technical or poetical, and because being technical seems
>>yet premature I will sum up by saying that with comp, the plenitude is just the
>>incredibly big "set" of universal machine's ignorance, and physics is the common
>>sharable border of that ignorance, and it has been confirmed because that
>>sharable border has been shown to obey to quantum laws.
>>I get recently new result: one confirm that with comp the first person can hardly know
>>or even just believe in comp; the other (related to an error in my thesis I talked
>>about in some previous post) is the apparition of a "new" quantum logic (I did
>>not command it!) and even (I must verify) an infinity of quantum logics between
>>the singular first person and the totally sharable classical discourses.
>>This could go along with your old theory that there could be a continuum of
>>person-point-of-view between the 1 and 3 person, and that would confirms that you
>>are rather gifted as an "introspecter" (do you remember? I thought you were silly).
>>But then it looks you don't like any more the 3-person discourse, why?
The adoption of the first person as a "frame of reference" (my terminology) implies the ultimate relativization. In other words, the logical system governing the mental processes of the observer becomes part of the "frame of reference> However, we all know that human beings do not think according to formal systems. Human systems are full of inconsistencies, errors, etc... and very often their beliefs about the world is just wrong. Very often they even make arithmetic errors such as 8x7 = 65.
So if we assume a relative formulation, here is the dilemma:
1) if we adopt a formal system such as the one(s) your have talked about we assign an absolute quality to the observer which violates our premise of relative formulation.
2) If we adopt a non-formal human logical system," we are left with an extremely complicated task of reconciling the observations obtained by several observers who in my terminology "share the same frame of reference"
One of the question that arise is how fundamental should be the concept of "frame of reference" or of the mechanism/logic that underlies our thinking:
1) Is it governed at the atomic level by physical laws down to resolution of Planck's constant? The notion of observer is defined here with a Planck resolution. If we share the same physical laws then we can say that we share the same frame of reference. This option avoids the inconsistencies of the "human logical systems" but throws out of the window the relativistic formulation. In addition this approach provides a neat justification for the equivalence of the sets describing the physical world and the mental world.
2) Is it governed at the neurological or even at the psychological level? The notion of observer here has a very coarse resolution compared to the first option. This approach keeps the relative formulation but becomes a quagmire because of its lack of formalism. How can the notion of "objective reality" be defined? In fact, is there such a thing as a true psychological objective reality? However, the fact that a "psychological objective reality" is an oxymoron (contradiction in terms) does not invalidate the definition of the observer at the psychological level. Au contraire.
Remember that my starting point is the computationalist hypothesis in the theoretical cognitive science. I take as objective truth arithmetical truth, and as third person objective communicable truth the provable arithmetical propositions like "1+1=2", "Prime(17)", or "the machine number i (in some enumeration) does not stop on input number j", this + Church Thesis + the "yes doctor" act of faith is what I mean by comp.
From this it will follow many things which can perhaps put some light on your questions and dilemmas, although, as you, see my point of departure is not a "relative formulation". What will happen is that physics will reemerge from what is invariant from all "relative point of view", which are themselves defined by the formal machines we are at some, necessarily unknowable, level. Indeed, in a second step, I interview the *sound* (by choice) universal machines on those invariant "through all relativities". The reasoning I invite people into occurs itselfs at a third person level, as do the interview of the machine.
But then, talking with the machine I need to (re)define some notion.
I (re)define science as the third person provability: thanks to Solovay this is formalizable by a modal logic G (+ that incredible G* which extends it at the "truth" level))
Let us write it simply by p. It means p is provable by me (me=the (hopefully) sound machine).
I define, following Theaetetus, the knowledge of p by the conjunction of p and p. That is "I know p" = p & p". Now the machine is sound, in particular the "truth theory" G* (the one I called the guardian angel sometimes) prove that
p is equivalent to p & p
So, from the *true* point of view: scientific provability and knowledge are equivalent. But, keep attention because here is the goedelian crux:
The sound machine itself does not, and cannot, prove or know that ( p is equivalent to p & p ). That is, the knower (or first person) defined by p & p
cannot know its "objective frame" from which p has been defined. The first person cannot know, neither proves, that she is any machine, although with comp
the machine can still infer the existence, or even bet on some presentation, of a machine through which he/she could hopefully survive.
This is important because although the knower and the "scientist machine" will know/prove the same arithmetical propositions, the logic of those
knowable, respectively provable, propositions differs considerably. "p" obeys to G (and G*), "p & p" obeys to the time/consciousness logic S4Grz.
G describes a sort of buddhist heraclitean (irreflexive) path where you can die, dream, get things wrong (like 8x7 = 65) at each instant, but S4Grz
describes ever evolving certainty-knowledge states.
(Do you see why the sound machine cannot prove that ( p is equivalent to p & p ) ? Because if the machine proves that, then the machine
will prove that p -> p, in particular the machine will prove false -> false, that is the machine will prove NOT  false, so the machine will prove her
own consistency, which no sound machine can do by Godel's second incompleteness theorem.)
You see, I take the self-reference logic as a sort of "exact third person psychology/theology". It cannot be normative because we cannot know
ourselves as consistent machine, and thanks to the difference of behavior between p and p & p, there is room for subtle inside views of arithmetic.
For the laws of physics it is the G*-equivalence between p with the big nuance p & <>p which plays the main role; and which will correspond
to the observable invariant relative to the consistent state of the machine. (Although since recently S4Grz does say interesting things too, I realize)
I mean, all the relative aspects of reality are captured by point of views (modalities) from inside arithmetical truth, which I take as absolute.
It is counterintuitive because the inside views will appear bigger than the outside view (like in Alice in Wonderland, Yellow Submarine, etc.),
but logicians are used to such relativity of views. They traditionally handle them with "model theory", or, in some case like our's "modal logic".
So to answer precisely your first dilemma between (I quote you):
<< 1) if we adopt a formal system such as the one(s) your have talked about we assign an absolute quality to the observer which violates our premise of relative formulation.
2) If we adopt a non-formal human logical system," we are left with an extremely complicated task of reconciling the observations obtained by several observers who in my terminology "share the same frame of reference" >>
My answer is that we can take both. The formal p and the unformal p & p. They are the same, the guardian angel says. But the machine cannot know that, there is a necessary ignorance which must be taken account. It is good because the UDA did show that physics emerges from such an ignorance.
*We* can do that, because through comp we reason at the upper purely arithmetical and third person communicable level.
Mmmh ... I certainly should explain better why p is formal, and p & p is unformal. The fact is that p interprets the arithmetical beweisbar Godel's provability, so you can translate p in arithmetic, but to translate p & p you would need an arithmetical truth predicate which does not exist by Tarski (see the thesis for a rigorous argument). At the higher level of description of course p & p is formal. Yes, G and G* are so powerful as being able to "metaformalize" unformality!
Concerning your other dilemma:
<< 1) Is it governed at the atomic level by physical laws down to resolution of Planck's constant?
2) Is it governed at the neurological or even at the psychological level?" >>
We will never know that. Some will bet on low level (meaning saying NO to the doctor for a very long time), other will bet on high level (saying quickly YES to their doctor). In all case it will be at their risk and peril, forever undecided. The reasoning I propose, and its translation in arithmetic, does not depend on the choice of the level, only on its existence.
Now, obviously, observation and introspection will give strong *evidence* for some levels, but on that matter cautiousness will *always* be needed.
Note I was assuming comp throughout.
I hope I have not been too technical, and that this helps a bit, and also that you are not too much disappointed that my approach relies so heavily and quasi-exclusively on the insane belief in the third person communicability of elementary arithmetic, but I know you knew that :)