Re: All feedback appreciated - An introduction to Algebraic Physics

From: Bruno Marchal <>
Date: Wed, 7 May 2008 15:40:00 +0200

Dear Günther,

Le 06-mai-08, à 21:19, Günther Greindl a écrit :

> - CRH implies COMP
> - COMP implies the negation of CRH
>> Universality, Sigma_1 completeness, m-completness, creativity
>> (in Post sense), all those equivalent notion makes sense only through
>> complementary notion which are strictly sepaking more complex (non RE,
> productive, ...).
>> The self-introspecting universal machine can hardly miss the inference
> of such "realities",
>> and once she distinguishes the 1, 1-plural, 3-person points of view,
> she has to bet on
>> the role of the non computable realities (even too much getting not
> just randomness,
>> like QM, but an hard to compute set of anomalous stories
>> (white rabbits, coherent but inconsistent dreams).
> Why does the machine have to bet on these complementary non-RE
> histories? I do not quite see how this arises from 1 and 1-plural POV;
> after all, it could be just rec. enumerable continuations?

Hmmm.... The UDA should just show that, and I am not sure which points
you are missing. Suppose there is a physical concrete universe and that
a UD is running in it without ever stopping (like in step seven of
Now, if you agree with the preceding steps, it should be clear that
your personal immediate future first person experience is determined by
the set of all computations going through your actual state. I agree it
is reasonable to consider such computations as equivalent with
recursively enumerable sets, but this does not mean that the set of
such continuations/computations is recursively enumerable. Actually,
such set is not even enumerable, because the UD, stupidly enough,
dovetails on some product of each computation with the real (oracle),
even the non computable reals. And, given that from a first person
point of view we cannot be aware of the infinitely many delays, we
have to take into account, to eliminate white rabbits (or to refute
comp) all the possible computations at once, including those
dovetailing on the reals or on any non enumerable structure. I have
discussed this with Schmidhuber on the list some time ago. The non
enumerability of the reals cannot prevent the UD to dovetail on all the
reals. From a third person point of view, everything is and remains
enumerable, but from the first person point of view, the subjective
indeterminacy has as domain something vastly bigger.

> Schmidhuber's/Tegmarks Computable Universe Hypothesis seems very
> attractive: it gives rise to structure, _evolves_ self-aware
> sub-structures, and gives reasonable (?) measure, for instance
> Schmidhuber's speed prior.
> This also takes care of the white rabbit.

Any constructive version of Feynman Integral will take care of the
third person white rabbits, but not of the first person white rabbits.
When we dream, although we loose soundness (depart from truth) we
remain consistent. The problem with comp is that there are a priori to
many dreams, too many consistent extensions ... unless *physical
reality* emerges from some gluing of *all* the dreams. I am just
arguing that for comp to be true, you have to extract a theory of
matter/body from first person coherence notions, eventually reducible
to notions from computer science/mathematical logic.

>> It's a bit like "understanding" (putting in a RE set) the (code of)
>> the total computable
>> functions, forces us to accept the existence of only partially
> computable functions,
>> which sometimes (most of the time, see the thesis by Terwijn) have a
> non recursive domain.
>> OK, the ontic part of a comp TOE can be no *more* than Sigma_1
>> complete,
>> but a non self-computable part of Arithmetical truth and analytical
> truth,
>> is needed to get the *internal* measure, we can't even give a name
>> to our first person plenitude and things like that.
> I think this answers part of my question above. The ontic part is only
> the Sigma_1 complete stuff; we assume the others for our measure ...

We don't assume the others, we are just confronted to them, like we are
confronted to the non stopping programs when we try to capture all
stopping programs (successfully, with Church thesis). The very notion
of Sigma_1 completeness forces us to accept machine Pi_1 incompleteness
(and then machine Sigma_2 incompleteness, etc...). And then the UDA
shows that our 1-indeterminacy domain (hopefully plural) is determined
by all that non computable stuff.
In the Whashington-Moscow self-duplication experiment, everything is
computable from a third person view, yet you cannot give an algorithm
predicting with certainty what precise experience you will feel to have
in your immediate future when you undergo it. Why do you expect a
computable reality when confronted with the whole deployment of the UD,
where you are reconstituted infinitely often in all consistent

> ... but
> my claim is that they do not give rise to first person experience.

By assumption (comp), first person experiences rise from some amount of
computations. But then the UD shows that you cannot know in any way
which computations generate your mind, and that your 1-expectations is
determined by all such computations. Now, empirically our expectations
are given by "physical laws". But if that is true, and if comp is true
too, those two ways to infer realities have to fit, so we have to
derive the physical laws from a sort of sum/mean/average-calculation on
*all* computations (even those with non computable oracle).
What I try to explain is that to solve the mind body problem, with comp
you have to justify the apparent computability of the observable
reality. You have to reduce somehow the body (physics) to the mind
(which, assuming comp is more easy: it is mainly computer
science/mathematical logic).

> I think the central question is this: _what_ does the Arithemetic Truth
> of whatever simulate? Reality at a granular level (like in the CA
> approach, Zuse's Rechnender Raum) - that is what I would assume - that
> reality at the lowest level is a number-relation; but that awareness
> only arises in these domains as a higher oder abstraction.

I can agree with this. The devil is in the details. Well, things get
more complex when you realize that the ontic has to be universal, and
that the physical (first person plural) has to emerge from all
computations. Each computation is computable (of course), but the
problem is in the *all*.
Even between two discrete steps of any computation going through your
actual experience, there will be infinitely many computations going
through those steps.

> I think you assume that the Sigma_1 sentences give the OMs directly,
> ....

Yes, in the sense that the UD generates succesively all 3-OMs.
No, in the sense that, by being unaware of any delays raised by the UD,
our 1-OM are never generated. We can only experience them, personnally,
and they depends on the whole structure bearing on *all* 1- OMs.

> is
> that correct? So in your view there is no underlying reality; QM and
> stuff like that is only an "illusion". Am I correct in how I interpret
> your theory?

Well yes ok, with the proviso repeated just above. And then a
vocabulary remark. In general "illusion" is a bit a pejorative. I would
prefer to say that QM and stuff like that are "emerging" . Given that
dreams and illusions are related to complex (uncomputable) sets of
computations, they obey rational principles, and, from a deep and
important perspective, they kick back and are not all illusionary. But
you are right, physics emerge from dream multiplication in the
universal deployment.

>> Perhaps this is why the Intelligible has been discovered (Plato)
>> before the "ONE"
>> (Plotin). It is far bigger. With comp you can restrict the ontic to
> the Universal Machine
>> (the baby ONE),
> Ok, I'm with you this far.
>> but its intelligible realm is well beyond its grasp.
> For me, the intelligible can be only a (proper?) subset of the ontic.

Ok, this is a bit subtle. I have been wrong myself often on things like
that. I mention one example: the set of total computable functions is a
proper part of the set of all (strictly partial and strictly total)
computable functions. But the set of total computable functions (TOT)
is far more complex than the set of partial computable functions (PFC).
  PFC is Sigma_0, TOT is Pi_2.
A simpler example: nothing is more simple than a dense rectangle. The
Mandelbrot set if far more complex. yet, the Mandelbrot set is a proper
part of that rectangle. A subset can be more complex that a superset.
Another example; Everett universal wave is very simple (not only
computable but linear!), yet it generates branches with observers
confronted to the ten thousand non linear and even non computable
Yet another example: there is no program capable of generating Chaïtin
Omega number, and generating *only* Chaïtin Omega number; yet, the UD,
or even simpler programs, generates it easily ... *among* all the
binary sequences.

> How could something that does not exist (ontic) be intelligible?

James Watson, a reductionist, says that only atoms exist. For him
molecules are already a construct of the mind. Yet molecules are
certainly intelligible for him (given that he has discovered the double
helix DNA).
Does a Bridge Play exist? Does a nation exist?
I prefer to say, contra Watson, that molecules, Bridge Plays, and
Nations, and People exist. The doubt is about the bottom, and I need no
more than numbers with addition and multiplication to explain how
stable and lawful illusions *exist* in the emerging and epistemic
sense. But then this move predicts that if I look at the bottom, and if
I want to say precise things about it (with all "decimals" correct),
then I have to take into account almost everything, both ontic and
epistemic (to be short).

> Or
> would you say that this is mathematical imagination?

If you agree that imagination exist, or at least follows laws.

>> All this is related to the fact, already understood by Judson Webb,
>> that comp is truly a vaccine against reductionist theories of the
>> mind.
> I have the Webb book on my desk and have glanced occasionally inside,
> it
> looks like a wonderful book, but I have not yet had the time to study
> it in detail.
> But I wonder - why do you say that comp is not reductionist?

In Philosophy of mind the term "reductionist" is often used for the
reduction of mind to matter. Comp forbids this (assuming com, assuming
I am correct, etc.). Of course you are left over with a reduction of
matter (physics) to mind (computer science/mathematical logic).
Comp is also non reductionist in insisting that "saying yes to the
doctor" has to be an act of faith, which eventually has to be a purely
personal decision. If you bet on comp, and if you "understand" comp,
you know you cannot be proselyte about it. It *is* a personal
"religion", despite its pure scientific consequences which can only be
used to refute it, never to make it a definitive statement. (I know we
will be in trouble when we will practice comp because some of us will
take their own survival as some kind of proof of comp, but personal
survival is (scientifically) never communcable.

> For me comp
> is reductionist - mind as the working of computation (I am pro
> reductionist, that is not a negative word in my view).

No problem as far as you see that computation has to be used in a
mathematical sense, contra Landauer and Deutsch who believes that
computation has to be reduced to the natural science.

> So, two questions:
> 1) At what level do your Sigma_1 sentence operate? OM's directly (I
> would interpret your paper in this way) or low level (more like a
> classical physical/digital physics view)?

This is a bit ambiguous. The true (and thus provable by universal
theorem prover machine (universal for computability, not provability,
but Sigma_1-provability universal) Sigma_1 sentences can correspond to
the 3-OMs, or to the "mental state" generated by the UD. But their
relative uncertainty first person measure (cf RSSA) are given by the
density (for some topology I try to derive with the lobian interview)
of their proofs-computation (those things are isomorphic because we
restrict ourselves to the Sigma_1 sentences).

> 2) You say that the ontic part is computable (in this sense, I would
> say
> COMP does _not_ refute CRH?) Because what "is" that is not ontic?
> That
> would be contradiction in terms?

Well, this is related to the (rather hot) debate on "emergence". I
have a Pythagorean-Kroneckerian slogan. All what exist "ontically" are
the natural numbers, together with their additive and multiplicative
structures. All the rest, including physics, exists as well, but not
ontically. It emerges epistemically from the way the intrinsical
ignorance of numbers about numbers (and sequences of numbers, etc.)
structure itself.

Now, that "intrinsical ignorance" is (with comp) what Godel has
discovered, and others (Lob, Solovay, + Post, Church, Turing ...) have
seen its structure; rather well captured by the modal logic G and G*,
and especially G* \minus G. That led to the Lobian interview ...

Please ask any questions. Be sure you have completely grasp the first
person comp indeterminacy before anything else (but the 1-3 distinction
of course). I do not pretend that all this is easy stuff, nor that I am
always clear about it. The problem is that what is simple for some is
difficult for others and vice versa.



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Received on Wed May 07 2008 - 09:40:07 PDT

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