On 17 Jan 2011, at 22:12, Evgenii Rudnyi wrote:
on 17.01.2011 14:00 Bruno Marchal said the following:
On 16 Jan 2011, at 22:27, Evgenii Rudnyi wrote:
...
Have you meant that the Universal Dovetailer will act for such a
situation according to Poincaré recurrence?
The UD will do that an infinity of times, given that the Poincaré
recurrence is a computable process. But the physical laws are sum on
first person views, based on a continuum of histories, so to relate
thermodynamic to the UD is certainly not obvious at all. The UD is
just a way to provide the minimal third person ontology (the
'everything') needed when we assume mechanism, and its role is to
build a mathematical formulation of the mind-body problem (if only to
illustrate that science has not yet choose between Plato and
Aristotle).
Let me write down how I understand this. The 3rd person view gives
us a complete Universe of numbers and the 1st person view perceives
just a part of it. Is this correct?
Unfortunately it is a bit more complex. Let me try to explain, even if
I introduce simplification, which eventually are wrong. They can be
handled only by the math, which are counter-intuitive on this.
If you want you have the "ultimate" third person point of view, which
is, assuming comp, just arithmetical truth. That is all the truth that
you can write in the language of arithmetic, that is the true formula
build from classical logic (with the symbols "&", "v", "~", "->", but
also "E" (it exists) and "A" (for all), together with the arithmetical
symbols "+", "*", s (successor) and "0".
The semantics of arithmetic formula is rather simple, because we have
a good intuition of the natural numbers. The symbol "0" is interpreted
by the number zero. s(x) represent the x + 1, etc.
For example the semantics of AxAy(x + y = y + x) is given by its truth
condition in the usual structure (N, + *). The formula
"AxAy(x + y = y + x)" is true if it is the case that for all numbers n
and m it is the case that n + m = m + n. OK?
Arithmetical truth is the collection of all those true formula. It is
a highly undecidable set. It contains Fermat theorem, but this has
taken centuries of complex math to prove. We don't know if it contains
Goldbach's conjecture, nor Riemann hypothesis, etc.
This can play the role of your "complete Universe of numbers". It
plays the role of GOD, or the ONE, in the (toy?) theology of the
Löbian machine. It is a highly non effective and non constructive
system, beyond the ability of any machine, and even any machine + a
hierarchy of strong non effective oracle. Actually, such an all
encompassing notion of truth cannot even be defined in the
arithmetical language of any machine (by a theorem due to Tarski).
Now enter the (digital) Löbian machine. A Löbian machine is a
universal machine/number. Universal means that it can mimic any
computable process if you give it enough time and (memory)-space. It
does not mean that it can PROVE all true statement of arithmetic, nor
even define its own conception of truth. By incompleteness it proves
only a tiny part of arithmetical truth. But "Löbian" means that it can
prove its own universality, and so it can prove its own Gödelian
limitations. It is a machine which "knows" that it is ignorant, or
more exactly that it has to be ignorant if it is consistent.
The beliefs of that machine are still third person view, a priori. Its
beliefs can be modeled by its (Gödel) provability predicate, and this
is an arithmetical predicate, and it belongs to the language of that
machine. We have, for p and q arithmetical formula, that
1) if the machine proves p, then the machines proves Bp
2) the machine proves B(p -> q) -> (Bp -> Bq)
3) the machines proves Bp -> BBp
And 4) the Löb formula, the machine proves B(Bp -> p) -> Bp.
Bp is the third person view of the machine. Think about the WM self-
duplication. The machine can prove, given the protocol that she will
be at W and at M. She looks at herself or itself in a third person way
(trusting the doctor about the description of her body).
The first person view is a notion far more subtle. It includes
consciousness or knowledge, which contains an implicit reference to
truth, and this can be used to show that neither consciousness nor
knowledge, nor the very notion of first person can be defined by the
machine. Yet, assuming mechanism, we can meta-define it very
precisely, by introducing an operator linking the belief of a
proposition p (Bp) with a clause saying that it is the case that the
proposition p (is true): Bp & p.
Socrate was already aware of the difficulty to define knowledge, and
in the "Theaetetus" Plato defines knowledge by the true justified
opinion, that is here Bp & p. The machine cannot define it, but could
perhaps reason on the meta-definition, like you and me.
The machine does not prove Bp -> p, nor p -> Bp, so Bp and Bp & p will
obey different logics, having different semantics.
What I call the first person view correspond to the logic of Bp & p.
It is indeed a logic of knowledge, even of evolving, self-developing,
knowledge. It is richer than the logic of Bp in many respect, and it
corresponds to a branch of a splitting and fusing tree, with a
topology akin to the topology of the real or complex numbers (although
a lot of works remains to makes those statements more precise).
And then there is that Skolem-like phenomenon which makes that first
person view richer than any third person view available to the
machine. This is due to the relation between proof (Bp) and the non
definable truth (p). If we describe in a third person view the
(intensional) content of that knowledge, it is much greater than what
the machine can prove, or even that GOD can prove! This is why I say
that arithmetical truth, seen from inside, is greater than
arithmetical truth (seen as a set of third person descriptions). The
thought experiment shows that such truth includes the many contingent
facts, like being in W or being in M, that you, but no one else, can
"know" by self-localization, in time and space.
For example, you are conscious here and now, because some arithmetical
proposition, involving your "body" at the right level of comp-
substitution, happens to be true and provable. But only *you*, (the
non definable knower) can know this, and this introduces a whole
different perspective on arithmetical truth, as seen by you.
Then the observability is defined in a similar way by Bp & Dp (Dp =
~B ~p, Dp is the consistency of p, the fact that there is a world
where p is true). This will explain why you are not only "here and
now", but will most probably stay near "here and now". Again this
makes sense because, by incompleteness, Bp does not imply Dp, for the
machine. Bp -> Dp is true, but not provable, and this changes again
the machine's perspective, and the logic and the semantics available
to the machine. Bp & p leads to intuitionist logic (with topological
interpretations), and Bp & Dp leads to a quantum logic, when "p" is
restricted to the UD-accessible arithmetical propositions, with
(hopefully) an Hilbert/von Neumann algebraic sort of semantics (linear
algebra, quantum mechanics).
Then sensibility is given by Bp & Dp & p. (Reapplication of the
Theaetetus trick). This leads to a qualia logic, extending the quanta.
To be sure, the quanta appears only there, making them qualia, and
making the whole physical reality a first person plural construction.
There is no *primitive* physical universe, only a secondary physical
phenomenon.
To sum up, mechanism leads to a neutral monism, where the basic truth
is arithmetic, and the "views from inside" are given by modal variant
of the self-reference logic (G, the logic of Bp, and their intensional
variants Bp & p, Bp & Dp, etc.). The person are abstraction, and in a
sense "Bp & p" is not a machine. The relation with the truth makes it
non definable in machine or number language. And indeed, the thought
experiment will link the knowledge of the machine with infinities of
consistent extensions which play a role in the semantics of Bp & p.
The ontology is very simple: arithmetical truth
But the theology and epistemology of the numbers will be terribly
complex. If *we* are machine, the theology is beyond the whole of
math, not just beyond arithmetic.
This is a bit like a painting, think of Mona Lisa. The paper on which
it is painted is very simple (a rectangle), but the drawing shape of
it is quite complex. A subset a of a set b, can be much more complex
than the set b. Likewise, the set of total computable function is much
more complex than the set of partial computable function which
contains it, and that is why the arithmetical intuitionistic logic of
machine is more complex than the whole truth, even if none of them are
definable by the machine. You can think also to the Mandelbrot set: http://www.youtube.com/watch?v=9G6uO7ZHtK8
So, to make my point clearer, it is not correct to say that the first
person point of view is a part of the third person point of view. It
is an entirely different *perspective* and a deep enrichment of the
truth, obeying different kinds of logic. The difference is almost as
big as the difference between a book on the history of the humans, and
the fact of being that particular human, or the difference between an
iterated self-duplication resulting in 2^1000 persons, and the fact of
being one of them.
I hope this can help a bit. It is because the notion of first person
is intrinsically complex that I am using the modal logic of self-
reference to handle them. Also, with mechanism, we have no choice: we
have to take into account the self-limitation results.
Best,
Bruno
http://iridia.ulb.ac.be/~marchal/
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