Bruno
I have read your post maybe five or six times, my hair getting grayer
and grayer everytime. This subject is undoubtedly your profession and
you are an expert at it but I have a lot of trouble following you.
Nevertehless, I have a good feeling to my stomach that you appear to be
on the right track.
You seem to say that you begin with an absolute formulation but end up
with a relative one, maybe the ultimate relative one. Not only that ,
you appear to have solved the paradox of the apparent objective reality
in the context of the ultimate relative formulation. This is good. This
is what I was hoping for. I think that philosophically, the ultimate
relative formulation is the most satisfying one. But this is only my
opinion.
I cannot lead the way but I can be a critic or a friend like Salieri to
Mozart :). Let's me see if I can convince you to bridge the gap and
maybe take the relative formulation as a starting point. Like Socrates,
let me start with one question. How can you possibly know to begin with
this particular assumption:
>> 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.
George Levy
Bruno Marchal wrote:
Hi George,
At 15:33 03/06/04 0700, George Levy wrote:
Bruno,
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)
GL wrote
<< It may be that using the observer as starting points will
force
White Rabbits to be filtered out of the
<< observable world
BM wrote:
>>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 1person 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 + AristotleKripke 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
>>personpointofview 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 3person
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 nonformal 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 certaintyknowledge 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 selfreference 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 nonformal 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 quasiexclusively on the insane belief in the third person
communicability of elementary arithmetic, but I know you knew that
:)
Bruno
http://iridia.ulb.ac.be/~marchal/
