> On 21 Jun 2018, at 06:44, Brent Meeker <[email protected]> wrote: > > > > On 6/11/2018 8:32 AM, Bruno Marchal wrote: >> Hi Telmo, >> >> >>> On 11 Jun 2018, at 13:53, Telmo Menezes <[email protected]> wrote: >>> >>> Hi Bruno, >>> >>> Sorry for the delay, had a friend visiting. >> >> No problem. From tomorrow (Tuesday) to Friday, I have many oral exams (+ a >> conference in Nivelles, a city nearby). So take your time to comment and >> express the dissatisfaction. >> >> >> >> >>> >>>> Ah! Let me try to answer.Keep in mind that I assume elementary arithmetic >>>> and thus computations, etc. >>>> (I am not sure I need YD here, but it can help). >>>> >>>> >>>>> - Why does consciousness even exist? >>>> Consciousness is somehow the doubt between consistency and truth (<>p v p). >>>> >>>> All universal number self introspecting meet this, and it is felt as >>>> immediately obvious, and thus true, and undoubtable, yet non rationally >>>> justifiable, and even non definable. >>> I follow your reasoning, from one of your recent articles. This leaves >>> me dissatisfied, but if I try to verbalize this dissatisfaction I feel >>> stuck in a loop. Perhaps this illustrates your point. >> >> We might need to do some detour about what it would mean to explain >> consciousness, or matter. >> I might ask myself if you are not asking too much, perhaps. Eventually, >> something has to remain unexplainable for reason of self-consisteny. I >> suspect it will be just where our intuition of numbers or combinators, or of >> the distinction finite/infinite comes from (assuming mechanism), or just why >> we trust the doctor! >> >> >> >>>> It goes from the rough dissociated universal consciousness of Q to the >>>> elaborate self-consciousness of PA or ZF, or us. >>>> >>>> >>>> >>>> >>>> >>>>> Darwinism does not seem to require it. >>>> It does. When the machine opts for <>p in the doubt between p and <>p, if >>>> it let it go, in some sense, it transforms itself into a more speedy and >>>> more efficacious machine, with respect to its most probable history. >>>> So, consciousness brings a self-speedable ability, which is quite handy >>>> for self-moving being living in between a prey and a predator. >>> I'm not convinced. Consider a simple computer simulation where agents >>> are controlled by evolving rules. Agents can eat blue or red pills. >>> 90% of the time blue pills give them energy and red pills cause >>> damage. 10% of the time the opposite happens. It is not possible to >>> know before eating a pill. Let's say the rule system evolves to make >>> the agents always eat blue pills and never red pills. Most of the time >>> this helps the agents, precisely because it assumes the most probable >>> histories. This is a simplified version of the sort of "decisions" >>> that evolution makes, and I would say that it is reasonable to assume >>> that our own evolutionary story consists of the accumulation of a >>> great number of such decisions. I still don't see how consciousness >>> makes a difference in such a mechanism. >> The reason why consciousness makes the difference is not related to the >> environment, but is intrinsic to the machine itself. >> >> I am aware to be quick on this, but the reason is a bit mathematically >> involved, and again, depends crucially of a discovery made by Gödel, and >> exposed in his paper “the length of proof”. >> >> Gödel discovered the existence that if you have some essentially undecidable >> theory, like RA, PA, ZF, there are always undecidable sentences, like <>RA >> in RA, of <>ZF in ZF, etc, then if you add an undecidable sentence (in the >> theory T, say) to T, you get a theory which not only will prove infinitely >> more sentence than T, but that infinitely many proofs will be arbitrarily >> shorter in T+the undecidable sentence than the proof of it in T, making >> “somehow” T+the undecidable sentence much faster than T. >> >> Even if the added sentence is false, we get that speeding-up > > ?? What does it mean that it is false? I thought "true" was undefinable.
True about the machine M is not definable by the machine M, but can be defined by some cognitively richer machine. Arithmetical truth is definable in set theory, analysis, etc. Here true meant “satisfied by the standard model of arithmetic, i.e. the usual structure (N, 0, +, *). > Do you mean it contradicts some theorem of T? No. I mean false, not inconsistent. Take the sentence "PA proves '0=1’ ”. It is false, but by incompleteness you cannot prove it in PA. (PA cannot prove it). So, you can add the sentence “PA proves 0=1” to PA, and you still have a consistent (yet unsound) theory. And the speed-up will still apply. > But in that case it would make T+the undecidable (false) sentence speed up > the proof of every sentence. T + the undecidable sentence remains consistent. The arithmetical []f -> f is not provable, and indeed the modal []p -> p not a theorem of G. It is a theorem of G*, and only G*. > >> (even for interesting sentences as Eric Vandenbussche convinced me (He >> thought that this was false, but eventually he proved that statement true). >> >> Blum has got a similar result in computer science, and eventually Blum & >> Marquez characterised the spedable machine/set (he used the w_i instead of >> the phi_i), and he obtained the class of sub-creative set, which generalised >> the creative set (which correspond to the universal machine). >> This means that if you take a slow universal machine, like the Babbage >> Machine, and a very efficacious machine, like a super-quantum computer, then >> you can by make the Babbage machine more rapid than the quantum computer on >> *almost* all inputs (= all except a finite number of exceptions), and even >> arbitrarily more rapid. Of course the “almost” limit seriously the >> applicability of that theorem, but in arithmetic, and for the FPI, that can >> play a rôle. >> >> In particular, take a machine which observe itself, and as some >> inductive-inference ability. By Gödel, or G, the machine can prove that if >> she is consistent, then her consistency is not provable. The machine can >> also see that she never succeed in proving her consistency, and eventually >> link this with the fact that her consistency (<>t) is not provable. Then, >> the machine can guess that she is consistent, by the adductive >> inductive-inference ability, and she can transform itself in a new machine >> with “<>t” added as a new axiom. That machine will be (theoretically) more >> efficacious (with some practical drawback). She can easily prove that his >> “ancestor” is consistent (in one line: “see the new axiom!”), and can prove >> infinitely more theorem, and can prove old theorem with shorter proofs. And >> she can continue on the (constructive, and then non constructive) >> transfinite. > > Can you give some non-trivial example of this speed-up? No. It is a proof by diagnoalization. A constructive version exists of that theorem, admitting a finite number or errors, but the program obtained are hard to describe in a few line. > >> >> This does not mean that a conscious machine is necessarily more efficacious >> on all task, > > What is the added undecideable sentence implied by consciousness? “I am conscious”. Bruno > > Brent > >> due notably to those finite number of exception, but it can be used to argue >> that in the long run, that make the machine more efficacious. >> >> Your exemple above is a sort of particular counter-example, but it take into >> account a social changing environment. Here I suppose the environment fixed. >> But if the environment changed, it will be even more benefices to compute >> more rapidly, even to find more quickly that she is wrong about its theory >> about her environment. >> >> >> > > -- > You received this message because you are subscribed to the Google Groups > "Everything List" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected]. > To post to this group, send email to [email protected]. > Visit this group at https://groups.google.com/group/everything-list. > For more options, visit https://groups.google.com/d/optout. -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
