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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