Re: Question about PA and 1p

[Bruno Marchal]

On 16 Jan 2012, at 20:42, David Nyman wrote:
On 16 January 2012 18:08, Bruno Marchal <marc...@ulb.ac.be> wrote:

I do not need an extra God or observer of arithmetical truth, to  
interpret
some number relation as computations, because the numbers,  
relatively to
each other, already do that task. From their view, to believe that  
we need
some extra-interpreter, would be like to believe that if your own  
brain is
not observed by someone, it would not be conscious.

I'm unclear from the above - and indeed from the rest of your comments
- whether you are defining interpretation in a purely 3p way, or
whether you are implicitly placing it in a 1-p framework - e.g. where
you say above "From their view".  If you do indeed assume that numbers
can have such views, then I see why you would say that they "interpret
themselves", because adopting the 1p view is already to invoke a kind
of "emergence" of number-epistemology.  But such an emergence is still
only a manner of speaking from OUR point of view, in that I can
rephrase what you say above thus: "From their view, to believe that
THEY need some extra-interpreter..." without taking such a point of
view in any literal sense.  Are you saying that consciousness somehow
elevates number-epistemology into "strong emergence", such that their
point of view and self-interpretation become indistinguishable from my
own?

It seems to me that this follows from UDA1-8. If not, then arithmetic  
if full of immaterial zombies, given that those computations does  
exist in arithmetic, in the usual sense of "17 is prime" independently  
of me. Or you need to reify matter to singularize consciousness, but  
this is shown by the movie graph (UDA-8, MGA) to be a red herring type  
of move.
Number relations does implement computations, in the same sense that  
brains' physics implement computations, by MGA.
Now the 1p are related, not on any particular computations in the UD  
(or in arithmetic), but to all of them, making both matter and  
consciousness not Turing emulable, but still recoverable from the  
entire work of the UD (UD*) or from the whole arithmetical truth. The  
point of view of some numbers will not differ from yours, given that  
yours is given by infinitely many such numbers relations. OK?

Bruno

Bruno



David


On 16 Jan 2012, at 15:32, David Nyman wrote:

On 16 January 2012 10:04, Bruno Marchal <marc...@ulb.ac.be> wrote:

Actually you can define computation, even universal machine, by  
using
only
addition and multiplication. So universal machine exists in  
elementary
arithmetic in the same sense as in the existence of prime number.


That may be, but we were discussing interpretation.  As you say  
above:
"YOU can define computation, even universal machine, by using only
addition and multiplication" (my emphasis).


Not just ME. A tiny part of arithmetic can too. All universal  
numbers can do
that. No need of first person notion. All this can be shown in a 3p  
way.
Indeed, in arithmetic. Even without the induction axioms, so that  
we don't
need Löbian machine.
The existence of the UD for example, is a theorem of (Robinson)  
arithmetic.
Now, that kinds of truth are rather long and tedious to show. This  
was shown
mainly by Gödel in his 1931 paper (for rich Löbian theories). It  
is called
"arithmetization of meta-mathematics". I will try to explain the  
salt of it
without being too much technical below.




But this is surely, as you
are wont to say, too quick.  Firstly, in what sense can numbers in
simple arithmetical relation define THEMSELVES as computation, or
indeed as anything else than what they simply are?


Here you ask a more difficult question. Nevertheless it admits a  
positive
answer.




I think that the
ascription of "self-interpretation" to a bare ontology is  
superficial;
it conceals an implicit supplementary appeal to epistemology, and
indeed to a self.


But can define a notion of 3-self in arithmetic. Then to get the 1- 
self, we
go at the meta-level and combine it with the notion of arithmetical  
truth.
That notion is NOT definable in arithmetic, but that is a good thing,
because it will explain why the notion of first person, and of
consciousness, will not be definable by machine.





Hence it appears that some perspectival union of
epistemology and ontology is a prerequisite of interpretation.


OK. But the whole force of comp comes from the fact that you can  
define a
big part of that epistemology using only the elementary ontology.

Let us agree on what we mean by defining something in arithmetic  
(or in the
arithmetical language).

The arithmetical language is the first order (predicate) logic with
equality(=), so that it has the usual logical connectives (&, V, - 
>, ~ (and,
or, implies, not), and the quantifiers "E" and "A", (it exists and  
for all),
together with the special arithmetical symbols "0", "s" "+" and "*".

To illustrate an arithmetical definition, let me give you some  
definitions
of simple concepts.

We can define the arithmetical relation " x =< y" (x is less than  
or equal
to y).

Indeed x =< y if and only if
Ez(x+z = y)

We can define x < y (x is strictly less than y) by
Ez((x+z) + s(0) = y)

We can define (x divide y) by
Ez(x*z = y)

Now we can define (x is a prime number) by

 Az[ (x ≠ 1) and ((z divide x) -> ((z = 1) or (z = x))]

Which should be seen as a "macro" abbreviation of

Az(~(x = s(0)) & ((Ey(x*y = x) -> (z = 1) V (z = x)).

Now I tell you that we can define, exactly in that manner, the  
notion of
universal number, computations, proofs, etc.

In particular any proposition of the form phi_i(j) = k can be  
translated in
arithmetic. A famous predicate due to Kleene is used for that  
effect . A
universal number u can be defined by the relation
AxAy(phi_u(<x,y>) = phi_x(y)), with <x,y> being a computable  
bijection from
NXN to N.

Like metamathematics can be arithmetized, theoretical computer  
science can
be arithmetized.

The interpretation is not done by me, but by the true relation  
between the
numbers. 4 < 6 because it is true that Ez(s(s(s(s(0))))+z + s(0) =
s(s(s(s(s(s(0)))))) ). That is true.  Such a z exists, notably  z =  
s(0).

Likewize, assuming comp, the reason why you are conscious "here and  
now" is
that your relative computational state exists, together with the  
infinitely
many computations going through it.
Your consciousness is harder to tackle, because it will refer more
explicitly on that truth, like in the Bp & p Theatetical trick.

I do not need an extra God or observer of arithmetical truth, to  
interpret
some number relation as computations, because the numbers,  
relatively to
each other, already do that task. From their view, to believe that  
we need
some extra-interpreter, would be like to believe that if your own  
brain is
not observed by someone, it would not be conscious.

Let me say two or three words on the SELF.  Basically, it is very  
simple.
You don't need universal numbers, nor super rich environment. You  
need an
environment (machine, number) capable of duplicating, or  
concatenating piece
of code. I usually sing this: If D(x) gives the description of  
x(x), then
D(D) gives the description of DD. This belongs to the diagonalization
family, and can be used to proves the existence of programs (relative
numbers) capable of self-reproduction and self-reference with  
respect to
universal (or not) numbers. So, some numbers can interpret by  
themselves
some relative number relations (relative to some probable local  
universal
number) as a self-referential statement (like "I have two hands"),  
or even
"I am hungry", making them hope some action in the environment will  
lead
them in most satisfying relation with that possible environment. Such
numbers can understand UDA like you and me, and realize that the  
only way
that is possible, is by its local reality being stable relatively  
to the
infinity of computations going through its computational states at  
its
correct comp level and below.

Tell me if this helps. I use comp throughout, 'course.

Bruno






David


On 14 Jan 2012, at 18:51, David Nyman wrote:

On 14 January 2012 16:50, Stephen P. King  
<stephe...@charter.net> wrote:

The problem is that mathematics cannot represent matter other  
than by
invariance with respect to time, etc. absent an interpreter.



Sure, but do you mean to say that the interpreter must be  
physical?  I
don't see why.  And yet, as you say, the need for interpretation  
is
unavoidable.  Now, my understanding of Bruno, after some fairly  
close
questioning (which may still leave me confused, of course) is  
that the
elements of his arithmetical ontology are strictly limited to  
numbers
(or their equivalent) + addition and multiplication.  This emerged
during discussion of macroscopic compositional principles  
implicit in
the interpretation of micro-physical schemas; principles which are
rarely understood as being epistemological in nature.  Hence,  
strictly
speaking, even the ascription of the notion of computation to
arrangements of these bare arithmetical elements assumes further
compositional principles and therefore appeals to some  
supplementary
epistemological "interpretation".

In other words, any bare ontological schema, uninterpreted, is  
unable,
from its own unsupplemented resources, to actualise whatever
higher-level emergents may be implicit within it.  But what else  
could
deliver that interpretation/actualisation?  What could embody the
collapse of ontology and epistemology into a single actuality?   
Could
it be that interpretation is finally revealed only in the  
"conscious
merger" of these two polarities?




Actually you can define computation, even universal machine, by  
using
only
addition and multiplication. So universal machine exists in  
elementary
arithmetic in the same sense as in the existence of prime number.  
All the
"Bp " and "Dp" are pure arithmetical sentences. What cannot be  
defined is
Bp
& p, and we need to go out of the mind of the machine, and out of
arithmetic, to provide the meaning, and machines can do that too.  
So, in
arithmetic, you can find true statement about machine going  
outside of
arithmetic. It is here that we have to be careful of not doing  
Searle's
error of confusing levels, and that's why the epistemology  
internal in
arithmetic can be bigger than arithmetic. Arithmetic itself does  
not
"believe" in that epistemology, but it believes in numbers  
believing in
them. Whatever you believe in will not been automatically  
believed by
God,
but God will always believe that you do believe in them.

Bruno










David

Hi Bruno,

  You seem to not understand the role that the physical plays  
at all!
This
reminds me of an inversion of how most people cannot understand  
the way
that
math is "abstract" and have to work very hard to understand  
notions
like
"in
principle a coffee cup is the same as a doughnut".


On 1/14/2012 6:58 AM, Bruno Marchal wrote:


On 13 Jan 2012, at 18:24, Stephen P. King wrote:

Hi Bruno,

On 1/13/2012 4:38 AM, Bruno Marchal wrote:

Hi Stephen,

On 13 Jan 2012, at 00:58, Stephen P. King wrote:

Hi Bruno,

On 1/12/2012 1:01 PM, Bruno Marchal wrote:


On 11 Jan 2012, at 19:35, acw wrote:

On 1/11/2012 19:22, Stephen P. King wrote:

Hi,

I have a question. Does not the Tennenbaum Theorem prevent the  
concept
of first person plural from having a coherent meaning, since it  
seems
to
makes PA unique and singular? In other words, how can multiple  
copies
of
PA generate a plurality of first person since they would be an
equivalence class. It seems to me that the concept of plurality  
of 1p
requires a 3p to be coherent, but how does a 3p exist unless it  
is a 1p
in the PA sense?

Onward!

Stephen


My understanding of 1p plural is merely many 1p's sharing an  
apparent
3p
world. That 3p world may or may not be globally coherent (it is  
most
certainly locally coherent), and may or may not be computable,
typically
I
imagine it as being locally computed by an infinity of TMs,  
from the
1p.
At
least one coherent 3p foundation exists as the UD, but that's  
something
very
different from the universe a structural realist would believe  
in (for
example, 'this universe', or the MWI multiverse). So a coherent  
3p
foundation always exists, possibly an infinity of them. The  
parts (or
even
the whole) of the 3p foundation should be found within the UD.

As for PA's consciousness, I don't know, maybe Bruno can say a  
lot more
about this. My understanding of consciousness in Bruno's theory  
is that
an
OM(Observer Moment) corresponds to a Sigma-1 sentence.


You can ascribe a sort of local consciousness to the person  
living,
relatively to you, that Sigma_1 truth, but the person itself is  
really
related to all the proofs (in Platonia) of that sentences  
(roughly
speaking).


OK, but that requires that I have a justification for a belief in
Platonia.
The closest that I can get to Platonia is something like the  
class of
all
verified proofs (which supervenes on some form of physical  
process.)


You need just to believe that in the standard model of PA a  
sentence is
true
or false. I have not yet seen any book in math mentioning  
anything
physical
to define what that means.
*All* math papers you cited assume no less.


  I cannot understand how such an obvious concept is not  
understood,
even
the notion of universality assumes it. The point is that  
mathematical
statements require some form of physicality to be known and
communicated,


OK. But they does not need phyicality to be just true. That's the
point.


  Surely, but the truthfulness of a mathematical statement is
meaningless
without the possibility of physical implementation. One cannot  
even
know
of
it absent the possibility of the physical.



it just is the case that the sentence, model, recursive  
algorithm,
whatever
concept, etc. is independent of any particular form of physical
implementation but is not independent of all physical  
representations.


Of course it is. When you reason in PA you don't use any axiom
referring
to
physics. To say that you need a physical brain begs the  
question *and*
is
a
level-of-reasoning error.


  PA does need to have any axioms that refer to physics. The  
fact that
PA
is inferred from patterns of chalk on a chalk board or patterns  
of ink
on
a
whiteboard or patterns of pixels on a computer monitor or  
patterns of
scratches in the dust or ... is sufficient to establish the  
truth of
what
I
am saying. If you remove the possibility of physical  
implementation you
also
remove the possibility of meaningfulness.



We cannot completely abstract away the role played by the  
physical
world.


That's what we do in math.


  Yes, but all the while the physical world is the substrate  
for our
patterns without which there is meaninglessness.






I simply cannot see how Sigma_1 sentences can interface with  
each other
such
that one can "know" anything about another absent some form of
physicality.


The "interfaces" and the relative implementations are defined  
using
addition
and multiplication only, like in Gödel's original paper. Then  
UDA shows
why
physicality is an emergent pattern in the mind of number, and  
why it
has
to
be like that if comp is true. AUDA shows how to make the  
derivation.


  No, you have only proven that the idea that the physicalist  
idea that
"mind is an epiphenomena" is false,


No. I show that the physical reality is not an ontological  
reality,
once
we
assume we are (even material) machine.


  And I agree, the physical is not a primitive in the existential
sense,
but neither is the information. Idealism would have us believe  
that
differences can somehow obtain without a means to make the  
distinction.



i.e. that material monism is false.


I insist everywhere that this is not what I showed. I show that  
all
form
of
weak materialism is incompatible with mechanism. All. The  
monist one,
the
dualist one, etc.


  How weak does materialism get when its primary quality is  
removed?
This
is a case of  "vanishing in the limit", something similar to  
the heap
that
vanishes when we remove the last grain.




A proof that I understand and agree with.


Clearly you did not. You even miss the enunciation of the result.
Mechanism
is incompatible with WEAK materialism, that is the idea that  
primitive
matter exist, or the idea that physics is the fundamental  
science.


  Can you not understand these words? How is materialism any  
weaker
than
the case of no material at all? My argument is that the  
possibility of
physical implementation cannot be removed without removing the
possibility
of meaningfulness. It is not an argument for a primitive  
ontological
status
for matter. You even seem to follow this reasoning when I ask  
you where
does
the computation occur then there is not paper tape for the TM  
and you
say
"on the walls of Platonia".



Your arguments and discussions in support of ideal monism and,


I prove that ideal monism is the only option, once you believe  
that
consciousness is invariant for digital functional substitution  
done at
some
level.


  No, you did not. Your result cannot do such a thing because you
cannot
have your cake (a meaningful set of expressions) and eat it too.
Digital
functional substitution is the substitution of one physical
implementation
for another, it shows that the fact of universality does not  
depend on
any
particular physical implementation but DOES NOT eliminate the  
need for
at
least one form of physical implementation. Digital  
substitutability is
an
invariance over the class of physical implementations, but what  
happens
then
you remove all members of a class? It vanishes!




like Berkeley's, still fail because while the physical is not
primitive,
it
is not merely the epiphenomena of the mind either.


It has to be by the UDA.


  And the UDA (like the UD) must have some implementation, even  
though
the
particulars of that implementation are irrelevant.


You are perhaps confused by the fact that unlike the physical,  
ideas
can
represent themselves.


I believe that comp makes the "physical" into an aspect of  
number's
self-reference.


  There we agree but I would say that a number's self-reference  
is its
connection to some physical representation. My point is that  
there
cannot
be
a self-reference without an implementation even if the  
particulars of
the
implementation do not matter.







If I take away all forms of physical means of communicating  
ideas, no
chalkboards, paper, computer screens, etc., how can ideas be  
possibly
communicated?


Because arithmetical truth contains all machine 'dreams",  
including
dreams
of chalkboards, papers, screens, etc. UDA has shown that a "real
paper",
or
& "real screen" is an emergent stable pattern supervening on  
infinities
of
computation, through a competition between all universal numbers
occurring
below our substitution level. You might try to tell me where in  
the
proof
you lost the arguement.


  When these "infinities of computations" are taken to have  
specific
properties merely because of their existence. You are conflating
existence
with property definiteness. Most people have this problem.


This does not make sense. I assume not just O, s(0), etc. I  
assume also
addition and multiplication. That's enough to get the properties.


  There is an "I" in that statement! What is this "I"? What is  
its
function? What class is it an invariant upon? Exactly how is it  
that
you
know of these properties? Absent the possibility of some form of
implementation in the physical, there is no distinction between  
you and
anything. Meaning requires distinction. Some even say that  
meaning *is*
distinction. What other than the persistence of pattern that the
physical
offers acts to allow for the ability to know differences?




Mere existence does not specify properties.


That's not correct. We can explain the property "being prime"  
from the
mere
existence of 0, s(0), s(s(0)), ... and the recursive laws of  
addition
and
multiplication.



  No, existence does not specify anything, much less that "0,  
s(0),
s(s(0)), ..." is distinct from any other string, nor does it  
specify
the
laws of addition or multiplication. Existence is not a property  
that an
object has.


Exactly. that's the point. You seem to contradict it.


  But existence is thus independent of properties and thus
distinctions.
So your claim that " "being prime" from the mere existence of  
0, s(0),
s(s(0)), ... and the recursive laws of addition and  
multiplication"
requires
a substrate that allows form representative patterns to obtain.
Universality
allows us to substitute one form of substrate for another so  
long as
the
function is the same. But universality and existence alone are
insufficient
for your claim that "I prove that ideal monism is the only  
option". You
also
have to show how the properties are both definite and  
invariant. This
requires implementation in a form that is invariant (to some  
degree)
with
respect to time. There is not time in Platonia therefore there  
in no
invariance with respect to time for the patterns of difference  
to occur
for
implementation to be said to obtain.





You need to study the "problem of universals" in philosophy, it  
is well
known and has been debated for even thousands of years. For  
example see
1
or
2.


This is a red herring.


  In a way, surely, but the essence of the problem is not. The  
paper
that
is reference 1 explains this well.





I go so far as considering that the wavefunction and its unitary
evolution
exists and it is a sufficiently universal "physical" process to
implement
the UD, but the UD as just the equivalent to Integers, nay,  
that I
cannot
believe in. “One cannot speak about whatever one cannot  
talk.” ~
Maturana
(1978, p. 49)


I think Maturana was alluding to Wittgenstein, and that  
sentence is
almost
as ridiculous as Damascius saying "one sentence about the  
ineffable is
one
sentence too much". But it is a deep meta-truth playing some  
role in
number's theology.


  OK, I deeply appreciate your erudition, you are much more  
educated
than
I am, but nevertheless, I submit to you that you cannot just  
ignore the
universals vs. nominal problem and posit by fiat that just  
because one
can
proof the truth of some statement that that statement's existence
determines
its properties. Our ability to communicate ideas follows from  
their
universality, that they do not require *some particular* physical
implementation, but that is not the same as requiring *no*  
physical
implementation. You argue that *no* physical implementation is
necessary;
I
disagree.


It is the result of the proof. It is up to you to show the  
flaw, or to
abandon comp.


  The problem is that mathematics cannot represent matter other  
than by
invariance with respect to time, etc. absent an interpreter.  
What you
seem
to think is that mathematics can prove things to itself in a  
manner
consistent with how I might be able to write out a set of  
symbols on
your
chalkboard that represent a proof of some theorem. You reject  
David
Deutsch's discussion of how this is wrongheaded out of hand,  
that is
unfortunate since it would greatly strengthen your case if you  
could
show
exactly where Deutsch is going wrong, if he is...




But I think that you  cannot define the universal wave without
postulating
arithmetical realism. In fact real number+trigonometrical  
function is a
stronger form of realism than arithmetical realism. Adding  
"physical"
in
front of it adds nothing but a magical notion of primary  
substance.
Epistemologically it is a form of treachery, by UDA, it singles  
out a
universal number and postulate it is real, when comp explains  
precisely
that
such a move cannot work.


  I am allowing for realism, it is a belief that may be true,  
but it is
not a unique singleton in the universe of models. I am arguing  
against
the
idea that the physical is primitive, against substantivalism  
especially
as
it is occurring in physics, for example see:
www.dur.ac.uk/nick.zangwill/Haeccieties.doc or 4.
  In physics there is a huge debate over the haecceity of space- 
time
and
your result is important in this, but your attempt to argue  
from the
other
side is as treacherous because it ignores the necessity of the
physical.


Comp makes necessary that there is no *primitive* physicalness.  
But as
David
points in his reply, you cannot say that I ignore the physical.  
The
whole
work is an explanation of why we believe in the physical, why  
and how
such
belief emerges and are persistent, etc. Physics is entirely  
given by
the
material hypostases, which are defined by number's self- 
reference, as
UDA
shows it to be the case necessarily so.


  This is insufficient. Merely postulating a property does not  
make it
so.
You continued intransigence on the non-existence of the  
physical world
with
statements that is shown to not be primitive is an avoidance of  
the
problem
by ignoring it, not a solution to it. The fact that is removing  
all
possibility of physical implementation by a theory of  
Everything makes
it
worse than mute, it eliminates itself as a meaningful theory  
simply
because,
to be consistent, it cannot be communicated.

Onward!

Stephen

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