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.
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.
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.
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).
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.
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.
Bruno
http://iridia.ulb.ac.be/~marchal/
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