On 22 Nov 2011, at 10:01, Pierz wrote:

OK, at last some time to sit down and reply properly. I want to come
back on this point about measuring proportions of an infinite set -
the measure theory you speak of. Now it seems clear enough that to
measure such proportions (say, the proportion of even numbers in the
set of natural numbers) one needs to iterate through that set in a
specific order. If one uses the counting algorithm n=n+1 iteratively,
then the result will be 50%, but if you use some other algorithm such
as the alternative one I provided, you get a completely different

You agree with this?

Not really. There are no uniform sigma-additive measure on N, or on discrete infinite spaces, but you can weaken the notion of sigma- additivity to simple additivity, and in that case there are solutions. See "amenable group" in wikipedia, for a summary on how to get rather nice, even uniform, "measure" on infinite discrete group.

Now, in the UD*, the measure does not bear on an infinite discrete space but on a continuum, because the UD, notably, reiterate infinitely self-duplications (like the little Mandelbrot sets do on their neighborhoods). The measure on first person consistent extensions are thus defined on a continuum, due to the first person invariance for the UD delays.

And the measure depends, and is even defined, by the geometry of the extensions, structured by the logic corresponding to the first person points of view. That is the part technically handled (even if only embyronically) in the "interview" of the LUM (AUDA).

Now this is an issue for UDA (it seems), because in order to calculate
the proportion of calculations in the infinite set in which I become a
giraffe, then we must iterate through those calculations in a specific
order. Otherwise, by arranging things the right way, I can get *any
result I want*. I demonstrated this in my post by showing how there
are more natural numbers divisible by a million than by 2.

Again, agreed?

The first person invariance results shows that the order of the states in the UD does not matter at all. What matter is the logical (including the epistemological) relationships that a state can have with the infinitely many universal machines going through that state.

OK, so I assume the order of calculations used to determine the
measure on the set must be the order they run in the UD.

Not at all. All what will count is a mix of redundancy, depth, and the self-reference constraints.

But my point
is that this order is *arbitrary*. This is because wherever the UD
uses a natural number n in its calculation, I can imagine some other
UD that uses someFunction(n) instead, where someFunction() transforms
n in such a way that all natural numbers are generated, but in a
different sequence.
There are infinite such alternative UDs. So why
should your UD algorithm be the 'real' one, simply because it uses the
limiting case where someFunction(n) is the identity function (return

Each UD generates all possible UDs.The "theology of machines", including physics, does not depend on the choice of any reasonable UD. Physics does not depend either of the precise ontology, as far as it is sigma_1 complete (emulate the UD).

It seems fatal to me - unless some other less arbitrary means of
counting the algorithms is (implicitly) employed. I say implicitly
since what I have read of the UDA from you seems to pass over this
critical question in silence.

I think I do the exact contrary. UDA exposes the problem, which is passed over by scientists since the neoplatonist have been banished from Occident in 500 and in Orient in the eleventh century.

AUDA illustrates the solution, by taking the machine points of view into consideration (as made obligatory by the mechanist mind body problem). It leads to a mathematical formulation of the mind-body problem, and to a theory of qualia and quanta satisfying the UDA requests.

I'd also like to put another question which relates to arithmetical
realism. Mechanism seems to be able to escape the UDA by denying
arithmetical realism in the first place - a doctrine which seems to me
to be far from self-evident, and certainly anathema to many

Arithmetical realism is the weaker hypothesis in all science, with the exception of ultrafinitist physicalism (an infinitesimal minority). Note that to define or assert that we are ultrafinitist physicalists, we need arithmetical realism. In fact: "NOT arithmetical realism" needs more than arithmetical realism. Someone really disbelieving AR should just say "I don't understand Pascal triangle", or "I don't understand all the fuss on the prime numbers", etc.

It is just the belief that the use of the excluded middle is sound for the first order logical sentences talking about the internal facts of the structure of (N, +, x). Intuitionists and classical mathematicians agree on AR, up to a change of vocabulary. Where a classic will say "p, but I have no constructive proof of p", an intuitionistic will say "not not p", basically. The real opposition is on the real numbers and on the sets, where comp is neutral, and take them as epistemological constructions a priori.

Then technically, the ontology uses only sigma_1 arithmetical realism (the idea that a digital machine either stop or does not stop). We can only "believe" in the excluded middle restricted to the sigma_1 arithmetical senetence (roughly: those having the shape ExP(x) with P decidable). Then the epistemology will already be bigger than anything conceivable. It is inexhaustible (it is really beyond math, with comp).

I have never met people who disbelieve in AR, except as a philosophical attempt to dismiss the mechanist questions when they begin to see the consequences. That's OK, but different, and rather natural given the lasting aristotelian prejudices.

The term "digital mechanism" has no meaning at all without arithmetical realism, nor does have Church thesis, or any part of computer science. Nor do the following terms: "periodic function, trigonometry, recurring phenomena, induction, anniversary, death, other people, etc".

Did you tell to your parents that your math teacher has gone mad the day he taught you integers and how to add and mulitiply them? If not you are enough arithmetical realist.

In the foundation of math, the goal is always to make sense of much bigger form of realism (on sets, categories, infinities, etc.) by the common sense we have on natural (finite) numbers.

I have, since sane04, eliminated the explicit AR axiom, because it is implicitly contained in Church thesis, and people tend to put too much metaphysical baggage in it.

On this matter I could cite Deutsch's claim that
computability is a function of the laws of physics, and that different
laws would permit different proofs and calculations, so to place the
computable functions prior to the physical world the way you have is
to put the cart before the horse.

UDA shows that Deutsch is wrong on this. Besides, the notion of computability is the only notion (with its relativized counterparts) which are close for the most transcendental mathematical operation (diagonalization). Deutsch uses an axiom to consolidate physicalism, without seeing that it contradicts its mechanist assumption in the cognitive science. The original Church thesis makes computability independent of theories and formalism.

We see a computable universe because
the laws of physics determines our brains as well as the structure of
the universe. This to me has a certain force to it, though no doubt
you will beg to differ.

It is not a question of private opinion. If you want the physical universe to be primary, you have to find a non computationalist theory of mind. You have to find a non Turing emulable relation between mind and matter. I am just studying the consequences of mechanism. I am showing the incompatibility of the conjunction of materialism and mechanism. I let people choosing the poison they prefer. This is the point of the UD argument. Of course AUDA shows that a priori, mechanism respects more the facts than materialism (which is rarely used in physics, except probably as a metaphysical background for not asking taboo questions on consciousness and mind).

BTW I disagree that I fail to understand the relation of 1-p and 3-p
in your proof.

You would not try to put some order on the comp state to get a measure, I think. You would see that comp leads to no choice in the matter (with a free pun, here). I mean that matter is secondary to any (turing) universal notions.

I am not making the same argument as before about the
infinite static field, and I do appreciate that our states are
represented in infinite calculations in the UD trace and that these
calculations are very deep, necessarily.

They have a priori the measure of the continuum (non enumerable infinity).

I also see how from your
reasoning, we would see an Everett-like uncertainty in our future

And Everett-like many-worlds/computations below our substitution level, together with statistical interference, in our present state. Comp forces to generalize Everett attitude (with respect to the quantum wave) on the whole of the arithmetical reality.

I don't see that you have pointed out any particular
misunderstanding on my part, though I am open to you explaining
exactly where in my reasoning this failure is.

Thanks for your explanation of my great-grandfather's work. I'm afraid
my physics is that of the very well read layperson, so I've never
really appreciated the ins and outs of what his contribution was -
other than "the statistical interpretation of quantum physics".

OK. That's another way to state "the Born Rules".

I read
the Einstein-Born letters too, many years ago, and enjoyed what I

Those are nice honest researchers, I think. Was Max Born still alive in 1957? I think so. Yes I see on the net he died in 1970. Did he ever knew about Everett? It would be nice to know what he would have thought about it. Bohr just rejects Everett abruptly.


On Nov 19, 8:49 pm, Bruno Marchal <marc...@ulb.ac.be> wrote:
On 19 Nov 2011, at 03:02, Pierz wrote:

In a previous post I launched a kamizake assault on UDA which was
justly cut to shreds on the basis of a number of misunderstandings on
my part, perhaps most crucially my conflation of information and
computation. I claimed that the UD cannot be distinguished from the
set of all possible information states and therefore from an infinite
field of static, within which all possible realities can be found,
none of which, however, have the slightest coherence. I also
mistakenly used the word 'random' to describe this bit field, which of
course is wrong. I should instead have used the word 'incoherent'.
Bruno and others quickly put me straight on these errors.

I am still troubled however by the suspicion that UDA, by explaining
'everything' (except itself - there is always that lacuna in any
explanatory framework) also explains nothing.

The UD is not proposed as an explanation per se. On the contrary UDA
shows that it is a problem we met when we assume that the brain (or
generalized brain) is Turing emulable.

Because the UD executes
every computation, it cannot explain why certain computations (say
Schroedinger's equation, or those of general relativity) are preferred
within our presenting reality.

That is basically my critics of Schmidhuber I have made on this list.

I'm afraid that you miss the role of the first person indeterminacy.
I will add explanation here asap. You have to follow UDA step by step:
it is a proof (in the theory "mechanism"), so to refute UDA you have
to say where it goes wrong. I insist: UDA is a problem, not a
solution. Indeed it is a subproblem of the mind-body problem in the
mechanist theory.
AUDA will be the solution, or the embryo of the solution.

This very universality also insulates
it against disproof, since although it allows everything we see, it is
hard to conceive of something it would disallow.

Not at all. A priori it predicts everything *at once*. That is the
"white rabbit problem".  We don't see white rabbits, or everything at
once, so mechanism seems to be disproved by UDA. The point will be
that such a quick disprove does not work, and when we do the math we
see mechanism is not yet disproved, but that it predicts or explain
the quantum weirdness.

David Deutsch's idea
of a good explanation is one that closely matches the structure of the thing it describes, allowing for little variation. The vast variation
in the possible worlds where UDA could be invoked makes it a bad
explanation, in those terms.

You have just not (yet) understood the role of the 1/3 person pov
distinction in the reasoning. UDA shows that physics is determined by
a relative measure on computations. If this leads to predict that
electron weight one ton then mechanism is disproved. UDA shows that
physics is entirely reduce to computer science/number theory in a very
specific and unique way (modulo a variation on the arithmetical
definition of knowledge).

Of course the objection that nobody has yet found an application for
UDA, a concrete example of its usefulness, is more of an objection to
it as a scientific theory than a philosophical one.

UDA is a proof. Unless wrong, it is done. Asking for the use of the
UDA is like asking for the use of the theorem saying that no numbers n
and m are such that (n/m)^2 = 2.
UDA shows a fact to be true and that we have to live with it. UDA
shows that mechanism and materialism are (epistemologically)

Still, I believe
there is an argument against it at the philosophical level. The UDA
invokes the notion of probability in relation to 1-p states on the
basis of the "infinite union of all finite portions of the UD in which correct emulation occurs". Thus the indeterminacy of 1-p experience is
a function of the distribution of states within the observer’s
consistent histories. For instance, there’s a 20% chance of x
happening, if it happens within 20% of my consistent histories. Please
Bruno correct me if this is a misunderstanding.

No, here I mainly agree with you.

Now we know from QT there is a finite, if absurdly remote, probability of my turning into a giraffe in the next minute. So the UD, if not to contradict science as it stands, must allow this too. And indeed there is no reason for it not to, since there must be computational pathways
that lead from human to giraffe - a sort of deep version of the
morphing algorithms used in CGI - or a simple arbitrary transform. In fact there must be infinite such pathways leading to slight variations
on the giraffe theme, as well as to all other animals, inanimate
objects and so on - okay let’s leave out the inanimate objects since
they possess no consciousness as far as we know, therefore no 1-p

Of course, these pathways are an extreme minority compared to the ones in which I retain my present form, behaving as we would expect on the
basis of the past.

"Of course"?
No, what UDA shows is that it is not obvious, and that computer
science can show it false, and so refute mechanism. But the math shows
that such a refutation, if it exists, is not trivial at all, and the
logic of self-reference shows that we are led to absurdities, not
contradiction (yet), and the absurdities are quite similar to the
quantum weirdness that we can "observe" (non locality, indeterminacy,
many worlds/dreams/states, symmetry at the bottom, etc.)

But here’s where I see the problem. In a
mathematical platonia we cannot make such a statement. The notion of
probability within an infinite set is untenable.

On the contrary. Probability calculus and measure theory have been
invented to put measure on infinite spaces.

It is analogous to
expecting that a number selected at random from the set of natural
numbers is more likely to be divisible by 2 than by, say, a million.
This is only the case of the set is ordered to appear this way, eg
1,2,3,4... If we write the set thusly: 1, 1 million, 2 million, 3
million, 2, 4 million, 5  million, 6 million, 3, 7 million.... etc
then our expectation breaks down.

You can use the usual Lebesgue measure on the 

Think about the repeated self-duplication. It shows that self-
duplication is a Bernouilli experience, so that in the limit (which
define the uncertainty domain for the first person experience), we can
use the usual normal distribution based on e^(- x^2) with the
normalisation factor.

So if there are infinite pathways where I turn into a giraffe, as
there must be, there is no way for my 1-p experience to select
probabilistically among these pathways. I can no longer say, if the
set of calculation pathways is infinite, that giraffe transformation
occurs in, say .000000001% of them, or 5%, or 99% of them.

Yes, you can. The problem is that the UD does not just iterate self-
multiplication (random noise), but it mixes it in a highly non trivial
way with infinitely many computations.

This is not a problem for an Everett -type multiverse, in which the
universes are bound together by consistent physical laws which allow
one to speak of a proportion of universes in which event x occurs.
However, in a mathematical platonia where all possible calculations
occur, and nothing outside of them, there can be no such ordering

If the Everett idea works, and is the solution, (which has not yet
been completely proved) then the UDA conclusion is that the Everett
simultion in the UD wins the "measure battle", and we HAVE to justify
this from computer science alone.

It would mean that the quantum computation are statistically more
frequent than the non quantum computations. But this must be shown, or
we miss the explanation of the origin of the physical laws, together
with the distinction quanta/qualia that digital mechanism already
explained (by the Solovay split between truth and proof).

I believe this same principle can be used to show that the
calculations of the UD must be disorderly. Consider some calculation c
which employs number n. In the UD there will also be a calculation
which instead uses the number n+1, another which uses n+2 etc. There
will also be calculations in which the ordering of the natural numbers
is rearranged in arbitrary ways such as my example above. Instead of
using simple n, the calculation will employ someFunction(n), where
someFunction() transforms the number as per my example, i.e. (in

if n modulo 4 = 0
   return n
   return (n-1) * 1,000,000

Thus the UD cannot rely even on the ordering of natural numbers to
‘prefer’ certain calculations, since the set of variants such as the
above will be infinite, and overwhelm calculations involving simple n.

This shows that the extraction of physics from numbers is not an easy
task, but again, you have to take into account the non triviality of
the 1 and 3 pov relation, and of computer science and mathematical
self-reference (G, G*, S4Grz, etc.) Then the shadows of why quanta and
qualia already appear.

Recently Eric Vandenbush (a guy who solved the first open problem in
my thesis) has found an explanation why the UD leads necessarily to
complex numbers for the measure problem (that's new! but I have yet to
be entirely convinced).

Pierz, I insist that the UD is not proposed as a solution, but as a
problem for DM. It is shown to be  an unavoidable problem we have to
solve if we keep digital mechanism in cognitive science. I am open
that it will lead to a refutation of mechanism, but the contraidiction
has not yet appear, and on the contrary, what we get is a similar
"many-world" problem that the physicists encounter too. This confirms
Digital Mechanism (DM) instead of refuting it. ...

read more »

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