On 20 Dec 2010, at 17:15, Jason Resch wrote:

On Mon, Dec 20, 2010 at 6:07 AM, Bruno Marchal <marc...@ulb.ac.be> wrote:

On 20 Dec 2010, at 03:15, Jason Resch wrote:

On Wed, Dec 15, 2010 at 4:39 AM, Bruno Marchal <marc...@ulb.ac.be> wrote:

But then a digital machine cannot see the difference between its brain emulated by a physical device, of by the true existence of the proof of the Sigma_1 relation which exists independently of us in arithmetic. Some will argue that a physical universe is needed, but either they add a magic, non comp-emulable, relation between mind and matter, or if that relation is emulable, they just pick up a special universal number (the physical universe) or introduce an ad hoc physical supervenience thesis.

I think multiple realizability applies to mathematical objects as well. Arithmetic may be simple enough to support minds and explain what we see, but should we discount the possibility that more complex mathematical objects exist, or that they are valid substrates for consciousness? I think a computer existing in a mathematical universe performing computations is ultimately still representing mathematical relations. If this is true, does it makes the UDA less testable or formally definable?

Once a computer exists in any mathematical structure, it will exist in the UD* (the UD deployment). But only the UD deployment can be defined in a way which does not depend on any choice of mathematical theory to describe it. Now, the measure of consciousness will depend on all mathematical structure, even if the measure bears only on the UD*, given that the measure pertains of first person experiences which are necessarily non computational. That is why the distinction between 3-ontology is 1-epistemology is very important. The true metamathematics of numbers is beyond numbers. The true theology of persons is beyond persons.

But doesn't this change the relative proportions that exist for programs contributing to a mind, and therefore change the likelihood of what one might experience? For example, do you see any reason for a civilization to upload their minds onto computers? Would this not increase the likelihood that their future experiences extend into this new reality of their choosing? Why should we bother to do anything at all if our actions don't change the relative measures of different conscious experiences?

Our actions change our relative computations measure, yes. I don't see the link with the fact that computers can be said to exist in model of theories having large cardinals. My point was that those computer's state are still generated by the UD, and if the measure depends of those high cardinal, that is what we have to justified from comp, once we assume that comp is true.

I agree. But the consequence seems to be a big leap for many. "Seems" because the results are more ignored than criticized. The problem (for many) is that mechanism is used by materialists, but in fine mechanism is not compatible with materialism. Mechanism makes matter an emerging pattern from the elementary arithmetical truth seen from inside. That makes mechanism a testable hypothesis, and that can already explain many qualitative features of the observable worlds, like indeterminacy, non-locality, non- clonability of matter, and some more quantitative quantum tautologies.

I thought non-locality is solved with Everett's interpretation, or do you mean the appearance of non-locality?

*Quantum* non locality is solved in Everett, and made into an appearance, indeed. But here I was saying that such an appearance of non-locality is already a theorem of (classical) digital mechanism.

I think I see what you are saying now. Consciousness can leap through space or time when instantiated elsewhere.

Yes. Notably through UD-times, and UD-spaces, or UD-memory (technically any 'Blum complexity', that is, decidable relations pertaining on the 3-description of the computations).

Also, I am curious how mechanism accounts for the non-clonability of matter.

By UDA, any piece of observable matter is determined in totality only by an infinity of computations. That is why the physical reality is NOT Turing emulable, and not describable by anything finite. To copy exactly any piece of matter, you would need to copy the results of the entire running of the UD (and extract the first person plural perception from it). Only your first person experience can interact with such piece of matter, but your digital mind always makes a digital truncation of that reality. That truncation leads to copiable things, but there are always approximation of the "real physical reality", which is really an infinite sum of computations. That's the rough idea. Russell is correct, it is better to attach the mind to all the instantiation in the UD, and then consciousness is a differentiating flux emerging from the number relations. Observation = selection of infinities of universes/computations among an infinity of universes/ computations.

Okay. I had thought you meant conservation of mass/energy, rather than the infinite complexity of matter.

A key idea not well understood is the difference between proof/ belief and computation/emulation. I will send a post on this.

I look forward to this post.

Searle can emulate (compute) the brain of a chinese. But Searle will not understand and live the conscious experience of that chinese (Searle category error, already well analysed by Dennett and Hofstadter in Mind's I).

I think Searle's mistake (perhaps among many) is that he substitutes himself in the place of the computer's processor, but this is like saying the physics understands Chinese (because the laws of physics are processing all the interactions in the Chinese speaker's brain).

Yes. It is typical level confusion. In the quotation below it would be the same as saying that RA to the thinking of ZF when RA emulates ZF. But of course it is ZF which thinks, not RA.

Likewise, PA cannot prove (believe) in its own consistency, but PA can emulate/compute completely the proof by ZF that PA is consistent. There is just no reason that PA begin to believe in the axiom of ZF. PA can emulate ZF, like Searle can emulate the chinese guy, but they keep different beliefs.

Here RA = Robinson Arithmetic, PA = Peano Arithmetic, ZF = Zermelo- Fraenkel set theory, ZFC = ZF + axiom of choice, ZF+K = ZF + the axiom of existence of inaccessible cardinals.

Emulation/computation is a universal notion, independent of any formal apparatus needed to describe those computations. But belief/ proof is highly dependent of the system used. It is not because I can emulate Einstein's brain that "I" will have Einstein's beliefs. But I will have Einstein computability power. And also, by emulating Einstein's brain, I can have a genuine conversation with Einstein (not with myself).

Once universal, all machine can emulate any other universal machine, yet they will have different and non equivalent provability abilities, and believability abilities.

It is useful to compare (<) theories in term of the portion of arithmetical truth that they can prove.

RA < PA < ZF = ZFC < ZF+K

Note that ZF and ZFC have different beliefs on sets, but the same beliefs on numbers!
ZF+K knows much more about numbers than all the other theories.

RA is the only one not rich enough (in provability) to be Löbian, but PA, ZF, ZFC, ZF+K, are Lobian numbers, and RA can emulate all of them. The key point is that RA cannot believe in general what they say. RA cannot prove its own consistency, but PA can already prove that RA is consistent, and RA *can* prove that PA can prove that RA is consistent. But that does not help RA, except if it feels alone and want to talk with someone richer than itself.

Only computation has such a remarkable invariance for change of systems, and that is a consequence of Church thesis. There is no such invariance for provability power. All theories (Löbian numbers) grasp only a tiny part of the Arithmetical truth, and all grasp a different parts (except ZF and ZFC). But they all compute the same computable functions.

It's not clear to me the role provability plays. Is the simple access to information within the state of a mind not sufficient?

I am not sure I grasp what you mean by 'simple access to information'. The only simple thing is your consciousness, and this is "just" a simple but no that simple, natural bet of all Löbian machine or numbers that there is a reality. That bet is natural in the sense that it occurs for all such machine/numbers due to the G/G* splitting (or simply: incompleteness). I write it often "Dt?", which can be justified by Gödel's completeness theorem which shows that to believe in your own consistency (Dt, or ~Bf) is the same as to believe that your belief are satisfied (true) in some world. You can prove such completeness for first order theory/machine or sufficiently effective higher order logic/machine/theory/believer.

Now, when you look at the moon, the "seeing the moon" and the "interpreting that seeing as an evidence for the moon" need the collaboration and 'voting' of millions of neurons, and they manage to integrate their works in such a way that you don't have to think consciously that your body/neurons are doing a very hard work (but well elaborated by millions years of evolution/deep-computational- history).

The role of provability consists in defining a very simple notion of believers. RA believes that 0+1 = 1+0, but RA has not the cognitive ability to believe that for any x and y x + y = y + x. PA has it, and thus are more beliefs on numbers, and ZF still more. Of course we (humans) are full of circumstantial belief axioms, like water is good, fire is bad, etc. Then I limit myself on perfect ideal believers like PA and ZF, ... because they have already a very rich theology, and we can understand why they have a soul, why they confused it with the 3-describable believer, why the soul falls and generate the material view, etc. The key is that computation does not need any special formal system, but belief and provability does. I identify a believer with a set of axioms (his body), close or not (depending of what I want to explain) for the deduction rules (= life, local relative dynamics).

Of course all the dynamics are emulated statically by the computation done by RA. There is an implicit use of the notion of (arithmetical) truth to distinguish an emulation of a computation from a mere description of a computation (that is a tricky point needed for the notion of computational supervenience).

Computation can be seen as a particular form of provability: indeed the Sigma_1 provability (that is: the proof of sentences having the shape ExP(x) with P decidable). RA is Turing universal = Sigma_1 complete. But Löbian believers like PA, ZF, ZF+kappa are not just Sigma_1-complete, they belief (correctly) that they are universal. This gives them the needed cognition ability to let the 'universal consciousness of the abstract Löbian numbers' differentiates through them, and select their consistent histories.

Just remember that
PA cannot prove its own consistency
ZF can prove the consistency of PA
and PA can prove that ZF can prove the consistency of PA, but cannot deduce from this that she is consistent. That is: emulating a machine having some belief or knowledge (of the chinese language for example) does not make the emulator to have such beliefs. To confuse computability with provability is a form of Searle's error. As you say, it would give all beliefs to the physical laws, or to arithmetic. Arithmetic emulate all believers, but does not have all the beliefs or all the believers (actually it would make it directly inconsistent).


That is why, also, ontologically, it is absolutely undecidable if there is anything more than sigma_1 (turing accessible) arithmetical truth. All the other arithmetical truth can be believed or not by such or such reasoner. The UD emulate (like RA proves) all the (conscious) beliefs of all machines, including ZF, ZF+K, etc. Consciousness is related to those computation/emulation of beliefs, not to the computations themselves. In a sense, a machine or a brain is never conscious: a relative machine, or a relative brain, just correlate consciousness experience relatively to plausible computation.

> No. The running of a program does NOT create a mind. It just makes it possible for a mind to manifest itself relatively to you. > The mind is already related to the platonic relations between the numbers which exist in an infinity of exemplars in Platonia.

If a single program does not create a mind, how does an infinite number of programs in the UDA create one? Perhaps I am unclear what you mean by mind.

Russell has given the correct answer. Here by mind I mean the conscious first person mind. By UDA-8 (MGA), consciousness is not attached to the physical running of a computer, but is attached to the logical number-theoretical relations describing that computation ... and all similar (with respect to the relevant levels) computations which exist in Sigma_1 (computational) arithmetical truth (and which might bear on beliefs and proofs which extends far beyond the computable).

Of course this is a delicate point. The notion of "a single program" is ambiguous. If it is a concrete physical instantiation of a program, then with digital mechanism, but also already quantum mechanism, it is already unclear if we speak about real infinities of indistinguishable histories/computations or of something unique (by taking some quotient of some equivalence relation).

Consciousness is never created. Consciousness comes from the fact that universal numbers can develop true (relative) beliefs, and that such true beliefs appears to be stable with respect of infinities of shared computational histories. From our point of view this consciousness *seems* to be related to our bodies, but this is a deformation-from-inside. Programs only makes possible for some *content* of consciousness to be correlated with those histories, and with the content of consciousness as lived by entities with which we share computational histories. It is, and has to be, counterintuitive. From "outside arithmetical truth" physical realities are "just" the intersubjective correlation of infinities of universal numbers beliefs. That is why I can understand very well Rex's first person feeling that consciousness is fundamental or basic. But numbers explains that feeling can be justified by the numbers relations, and have to, if we accept the existence of the substitution level.

Hope this helps.



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