On 17 Jan 2011, at 22:12, Evgenii Rudnyi wrote:

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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 givesus a complete Universe of numbers and the 1st person view perceivesjust 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/ -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.