OK, well I think this and the other responses (notably Jason's) have
brought me a lot closer to grasping the essence of this argument. I
can see that the set of integers is also the set of all possible
information states, and that the difference between that and the
the element of sequential computation. I can also see that my
objection to infinite computational resources and state memory comes
from the 1-p perspective. For me, in the "physical" universe, any
computation is restricted by the laws of matter and must be embedded
in that matter. Now one of the fascinating revelations of the
computational approach to physics is the fact that a quantity such
position can only be defined to a certain level of precision by the
universe itself because the universe has finite informational
resources at its disposal. This was my objection to the UD. But I
see that this restriction need not necessarily apply at the 'higher'
p level of the UD's computations. What interests me is the question:
does UDA predict that the 1-p observer will see a universe with such
To be sure, this is an open problem.
To be sure, this is an open problem for physicists too.
Comp+"theatetus" will be refuted if the comp-physics will be quasi
*contradicted* by some precise physical fact, not by any physical
theory (unless they predicts that precise physical fact).
If it explains why the 1-p observer seems to exist in a
world where there is only a finite number of bits available, despite
existing in a machine with an infinite level of bit resolution, then
that would be a most interesting result. Otherwise, it seems to me
remain a problem for the theory, or at least a question in need of
answer, like dark matter in cosmology.
I am going to have to meditate further on arithmetical realism. I
don't believe in objective matter either (it seems refuted by Bell's
Theorem anyway), but a chasm seems to lie between the statement "17
is prime" and "the UDA (Robinson arithmetic) executes all possible
Don't confuse the UD (Universal Dovetailer, a finite program) and UDA
(the UD Argument = the argument that, assuming digital mechanism,
physics is in principle a branch of number theory/computer science,
and in which the UD plays the role of the effective definable comp
ontological realm of *everything*).
Just a vocabulary remark, to avoid possible future confusion.
The problem is one of instantiation. I can conceive of a
universe - a singularity perhaps, with only one bit of information -
in which the statement "17 is prime" can never be made.
Don't confuse the sociological statement "some machine asserts "17 is
prime"", and the true fact that 17 is prime, which does not rely on
physical universes at all, a priori.
ie instantiate, 17, requires a certain amount of information.
In some physical theory, but this is not an assumption in the theory.
You cannot refute an argument by adding new assumptions.
that a program executes, as opposed to saying it merely is implied
a set of theoretical axioms, requires the instantiation of that
In Aristotelian metaphysics.
Also, even in platonia, a computation is described by a big number
(possibly infinite) of implications.
I suppose this is a restatement of the problem above.
Arithemetical realism then would be the postulate that everything
implied in arithmetic is actually instantiated.
Not at all. That would be a physicalist revisionist definition of
numbers. You need to "instantiate" 17, in some way, to talk about 17,
but 17 itself does not need instantiation. With or without any
physical universe, 17 remain a prime number.
Now, an instanciation, or emulation, can be defined from the numbers
alone. Some numbers are universal (a relative arithmetical property)
and we can say that a universal numbers instantiates 17 (say) if 17
appears in some of its purely arithmetical register.
To understand the detail of this, I can only refer you to some good
textbook in computer science. The main theorem for this is the proof
that all partial recursive functions can be represented in Robinson
arithmetic (Boolos and Jeffrey's book do this very well, Epstein and
carnielli also. ref in my theses).
It seems to me I can
grant 17 is prime, without granting this instantiation of
Well, that solves you of a very long and not so easy work.
I'm also troubled by the statement that you have proved in the AUDA
that any Lobian machine can apprehend the UDA. Is not a three-year-
child and a cat a Lobian machine? Or indeed my senile father. How
you assert they could comprehend such an abstraction? Either they
aren't Lobian machines, or there's hole in the proof somewhere,
Recently I have updated my spectrum of Löbian machine to the octopus,
and the jumping spider. I can argue that they have the cognitive
ability to get UDA. But they don't have a sufficiently big brain to
exploit this, and they don't have the motivation to use diaries and
books, and language to generate their infinite "turing tape memory"
like we do.
Symptoms of Löbianity are believe in repetition and notice them (like
believing in a notion of anniversary), or having empathy for an other
creature, etc. This needs some form of the induction axiom. Robinson
arithmetic (and Universal machines in general) are not Löbian. Peano
Arithmetic is Löbian (it is reallu just Robinson arithmetic + the
induction axiom for the first order describable formula).
But this is not important for the reasoning.
Jason mentions the anthropic principle (which of course I'm well
acquainted with) and the idea of the computations which contain
observers. I have read, without following, some of your propositions
involving the Beweisbar predicate and self-referential relations and
what have you. Is that the formalism that is supposed to define
computations are conscious?
Not really. This is a subtle point. Notion like truth and
consciousness are not definable by any machine. But, like with God
Plotnus' one) we, the machines, can talk in indirect way, by taking
Is there a summary somewhere?
It is explained in the second part of the sane04 paper. AUDA is "the
interview of the Lôbian universal machine"
wondering how consciousness can possibly be an attribute of some
computations and not others,
Let me be precise; consciousness is not an attribute of a
but is an attribute of a person. Now a person can manifest itself
relatively to other person, once "enough" similar computations are
going through the states of the two person, in some sufficiently
cohesive way. The self-reference logics are used to single out the
conditions of cohesion (unlike in linear logic, or Girard geometry of
interaction, which extract such condition from symmetry intuition and
and why, if it's a matter of some certain
mathematical properties of the computations, we could not fairly
easily write a conscious algorithm?
It is easy. I tend to think, since recently, that all universal
algorithm are conscious. But their consciousness is disconnected, a
bit like if they were born ... in salvialand! And, yes, before doing
salvia I would have imposed Löbianity for consciousness, but I am
less sure about that.
Now Löbianity is more than consciousness, it is self-consciousness.
Peano arithmetic is self-conscious, I think. That is why we can
discuss Plotinian theology with them, even without making their soul
falling on earth, that is without implementing them and sharing our
long story. Current computers have not yet long term memory, nor long
term goal. But I think that PA, the octopus, and the jumping spider
(but not worms, and most usual spider) are as conscious as you and
For the fun here is video illustrating that a jumping spider can do
some inductive inference requiring some implicit beliefs in
arithmetical induction (look hw she reacts when she looks behind the
As opposed, here is a typical non Löbian behavior, or a non jumping
spider (yet jumping, note):
If not finding food on a top of a plant, she is programmed to jump
randomly on other plant, and, in case she get the ground, to climb
a nearest plant. Here there is only a pen, perpendicularly installed
on a flat ground table. She seems to repeat in cycle that behavior,
except for taking some rest.
But the bigger reason why I think jumping spiders are Löbian, is that
like cat and dog, they can bond with you, star at you, and perhaps
even recognize you. But this can be judged only by real interaction
with real spiders, not by looking at videoas, of course. Stiil, here
is a very cute one:
Surely complexity can't be the
defining feature (at what arbitrary level of complexity does the
come on?), so it should be a simple matter.
I agree. You don't need more than 10 lines instruction code for them,
well, in a high logical language like prolog, for example.
(Though the proof of
having created consciousness in the program would be a problem!)
It is not a problem. It is an impossibility. You cannot prove that
am conscious, can you?
we have to define consciousness (not necessarily self-awareness, or
the awareness of being aware) as a property of numbers per se?
A quasi-definition is the ability by some universal numbers to
discover some non communicable truth by introspection. Consciousness
is not much more than the state of believing in ...
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