# Re: Question about PA and 1p

```
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
```
```
```
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?
```
```
```
```

```
```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
```
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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