On 03 Apr 2011, at 05:15, stephenk wrote:


On Apr 1, 1:58 pm, Bruno Marchal <marc...@ulb.ac.be> wrote:
On 31 Mar 2011, at 20:16, Stephen Paul King wrote:

-----Original Message----- From: Bruno Marchal
Sent: Thursday, March 31, 2011 12:33 PM
To: everything-list@googlegroups.com
Subject: Re: IsQTIfalse?

On 31 Mar 2011, at 15:35, Stephen Paul King wrote:

snip

Hi Bruno,

  I understand the role of the infinities of computations and the
equivalence as you are considering them finally, from reading your
papers over and over and a brilliant discussion of the concept of
quantum superposition in Andrew Soltau's book Interactive Destiny,
but am still not seeing the conflation of physical causality and
logical entailment. For one thing they point in opposite directions!

Let us say that this is an open question in the comp physics. I
understand Pratt motivation, but imo, he simplifies too much the mind,
and abstract himself from the comp hyp. It might be that we have a
time relation A ===> B related to the "BD" definition involving A - > B.


[SPK] Forgive me, I don't know the definitions of these different
arrows. Pratt does speculate that there is a duration component
involved in interactions.

"It is ironic that Cartesian philosophy, whose guiding dictum was to
question
everything, should question causal interaction between the mental and
physical
planes before that within the planes. The latter problems must have
posed an
insufficient challenge to the Cartesians. We argue that the converse
is the case:
between is actually easier than within!
We interpret interaction as causality. Causality is directional, but
the direction depends on whether we have in mind physical or mental
causality. We interpret x |= a ambiguously as the time elapsed between
the occurrence of the
physical a and its impression on the mental state x, and as the truth
value of a as a proposition.
The former is physical causality or impression, flowing forward in
time from events to states. The latter is mental causality or
inference, flowing backwards in time from the thought of a to the
inference of a’s"

 His use of the word "causation" is unfortunate but we can forgive
him because there is no correct word for the relation that he is
considering. The idea if more analogous to the arrow of implication in
logic but in a physical context. Because of the linear superpositions
of QM we cannot think of causality as a strict bijection. It is
possible to derive the bijective aspects but we cannot start with
them. This is the key idea that Pratt is exploring!
 What one has to understand is that he is considering evolution of
both logical structures and their dual Stone spaces under a single
system, the Chu transform. All he is doing is taking the Stone
Representation theorem and the Pontryagin duality seriously that there
is a general duality between logical algebras and certain kinds of
spaces and if one allows for the possibility that the logical
structures can evolve then there would be a co-evolution in the dual
spaces, an evolution that looks exactly like that we are considering
in physics: particles moving around in space-time.

http://en.wikipedia.org/wiki/Stone%27s_representation_theorem_for_Boolean_algebras

"A simple version of Stone's representation theorem states that any
Boolean algebra B is isomorphic to the algebra of clopen subsets of
its Stone space S(B). The full statement of the theorem uses the
language of category theory; it states that there is a duality between
the category of Boolean algebras and the category of Stone spaces.
This duality means that in addition to the isomorphisms between
Boolean algebras and their Stone spaces, each homomorphism from a
Boolean algebra A to a Boolean algebra B corresponds in a natural way
to a continuous function from S(B) to S(A). In other words, there is a
contravariant functor that gives an equivalence between the
categories. This was an early example of a nontrivial duality of
categories.

The theorem is a special case of Stone duality, a more general
framework for dualities between topological spaces and partially
ordered sets."

from: 
http://golem.ph.utexas.edu/category/2008/11/variations_on_pontryagin_duali.html

"Pontryagin duality goes like this.  Suppose A is a locally compact
Hausdorff topological abelian group.  Let A * be the set of
characters: that is, continuous homomorphisms f:A→U(1). A * becomes
an abelian group thanks to pointwise multiplication of characters. It
becomes a topological group with the compact-open topology — that is,
the topology of uniform convergence on compact sets.  We call A * the
Pontryagin dual of A.

Then, A * is again a locally compact Hausdorff topological abelian
group, and

A **≅A

in a natural way!

For example, we have

ℤ *≅U(1)

and

U(1) *≅ℤ

ℝ is its own dual!  More generally, for any finite-dimensional real
vector space V with its usual topology, V * is the same as the dual
vector space.  So, Pontryagin duality generalizes vector space
duality."

 The Pontryagin duality extends Stone spaces such that they are
capable of exactly representing "particles" in that they are
"disconnected". Thus we have minds - as evolving logical structures -
and bodies - as fields of separate locally compact Hausdorff
topological groups. What connects them are the properties - what they
do.

I still don't understand how you persist in not seeing the
implications of the Stone duality!

Explain. I don't feel like missing it.


[SPK] That logical structures alone are insufficient to model our
existence.

Correct. But arithmetical structure are enough (or please mention a flaw in UDA).



We need the physical world to be the interface between our
separate minds,

Eventually with comp, the physical world is recovered by defining it as an interface between our different minds, or as the gluing dreams processes. We need a physical world. No doubt on this. The point is that we don't need a primary physical world.





otherwise we will be trapped in the UD in endless
Poincare recursions. This is the nightmare that Nietzsche saw.

I doubt this, but if that were true, that would not been a reason to abandon comp. Only a reason to hope that comp is false. But comp is not yet sufficiently developed to start having premature fear of it.






Oh well, that is your choice,

I am problem driven. I don't make choice.


[SPK] You are choosing to not consider multiple interacting minds.


Why do you say so? comp starts from the interaction between a patient and its doctor. Comp is an hypothesis. If it leads to solipsism, that would be a reason to abandon it, indeed. But everything points on the fact that there is all the room needed for mind interaction, and even that this is what stabilizes the first person plural in the long run.



So
far I have only seen discussions in your papers in terms of
"interviews" between different logics.

There is an 'interview' between a human (you and/or me) and a universal machine. The logics are related by representation theorems, as usual.



What you are calling
interviews, I would call them interpretations or mappings.

"interview" just means that I am in front of the machine, and I have to ask her about each different points of view. I just translate the usual classical theory of knowledge in terms that the machine can understand. So of course, we are lead to mappings and representations.



There is no
notion of separable entities having anything like what you and I are
doing right now here.

You are not at the right level. You could criticize string theory because it does not bring you a pizza at home tonight. The interaction comes from the linear combinatory algebra. But if I posit at the start, I will lose the qualia. I have to derive that linear algebra from the gluing property of the machine dreams (UDA shows we don't have choice in that matter). If eventually the machine dreams does not glue well enough, we will know that comp is false, with some degree.



You wrote brilliantly about your idea of
interviews here 
http://www.mail-archive.com/everything-list@googlegroups.com/msg08457.html
But I will continue to argue that "the logic of arithmetical self-
reference" is not an exchange of information between separate minds.

It is not supposed to be that. The logic of Bp & Dp should bring such a thing, or, if you can prove it prevents such exchanges, then comp +Theaetetus is refuted.



It is at most the exploration of 1p aspect of a logic by that logic.
It is solipsism at its most exquisite form. (Please understand that
this is not a bad thing, solipsism is thinking and dreaming about
one's thoughts in a closed and convex form).

It is not solipsism-the philosophy.
Bp & p, and Bp & Dp & p, leads to "lived solipsism", which is the case for the first person internal experiences. But the modality without "& p" are not solipsist at all. You are conflating different modalities.





But there is something else that troubles me even more.

 You wrote in http://iridia.ulb.ac.be/~marchal/publications/CiE2007/SIENA.pdf
:

"Each hypostase will be interpreted by a set of arithmetical
sentences.
Plotinus’ One is interpreted by Arithmetical Truth, i.e the set of all
true arithmetical sentences. In
case we were interviewing ZF, we would have needed the more complex
set-theoretical truth. In any
case, it follows from Tarski theorem that such a truth set is not
defineable by the machine on which
such truth bears. Nevertheless, she can already, but indirectly, point
to its truth set by some sequence
of approximations, and there is indeed a sense to say that Lobian
machines are able to prove their
own “Tarski theorem”, illustrating again the self-analysis power of
those theorem prover machines. See
Smullyan’s book [60] for a sketch of that proof and reference therein.
In this sense we recover the “One”
ineffability, and it is natural to consider arithmetical truth as the
(non-physical) cause and ultimate
reality of the arithmetical machine. This is even more appealing for a
neoplatonist, than just a platonist,
given the return of the neoplatonist to the Pythagorean roots of
platonism [52]. The atomical verifiable
“physical” proposition will be modelized by the Σ1 sentences. Note
that the machine can define the
restricted, computationalist, notion of Σ1-truth."

 The problem is that "the set of all true arithmetical sentences" is
a very narrow, but deep, interpretation of the One. How can I define
such things as Zeno's paradox and its solution, for example?

It is basically solved at the start, because real numbers are epistemological, or meta, construction. Comp suggests that the ontology is discrete, because we can explain the beliefs and uses of the rest from that.




There is
no way to define an infinitesimal or a derivative that I can find.

Because comp makes the real numbers a simplification, and it makes calculus a handy tool for manipulating big numbers and and epistemological mind constructs. Analysis and physics are epistemology. This follows from UDA + some amount of Occam.



How
do I recover the calculus?

In the stable numbers' dream.




Your model has no expressions that can be
used to act as a clock...


I told you that the definition of integers *is* a clock. Arithmetic starts from a clock. And besides, I have no model. Only a theory (that I am digitalizable at some level, yes doctor + CT).



Thus it is no surprise that the whole
structure is frozen.

The point is that after Gödel, nothing is more dynamical than Platonia, when seen by the creature defined internally in Platonia. If you assume a real fundamental time, you have just to abandon comp (and special relativity which makes time an illusion too). Anyway, time and space are things which I prefer to search an explanation for, than assuming them at the start.



There is no room in it for the idea of evolution,
nothing 'becomes".

When the UD is executed, all the becoming becomes. And so all possible evolutions develop. You could as well criticize SR and GR, and QM (without collapse). I mean, this is a place where comp already agree with most physicists, except Prigogine.



Everything just "is".

Only in God's eye.



Every fiber of my being
screams out in revulsion at this!

There is no reason, but apart from solipsism, we cannot use such affirmation as an argument. You could say that Energy is not equal to mc^2 because we can do horrible bombs with that idea.




I am not a Σ1 sentence!

I guess you mean: my mental state is not UD-accessible. Just say "no" to the digitalist surgeon, Stephen. I don't know if that is true or not. My point is that if it is true, then physics is a branch of number theory, and I show how time space and physics can indeed to be retrieve. There is already subjective duration, but not yet space.

We would disagree only if you want both
- a fundamental basic *primary* time, and
- saying yes to the doctor.
OK?




http://www.google.com/url?sa=t&source=web&cd=12&ved=0CGgQFjAL&url=http%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fdownload%3Fdoi%3D10.1.1.66.6645%26rep%3Drep1%26type%3Dpdf&rct=j&q=%22I%20am%20not%20a%20number%22&ei=QuWXTf2_CJCO0QGLquTkCw&usg=AFQjCNHUw_V2FSvbiVGXt-ivhgdbu55n4g&sig2=NwY4YJrHlExXauSu-ceiRQ&cad=rja

Despite the title of that paper ("I am not a number, I am a free variable"), it is quite coherent with comp, even philosophically close, given that the 1-I can be seen as the "free" places of your possible occurrences in a continuum of computations.






but putting that aside the continuity of 1st person should supervene
on the UD, no?

It is more correct to say that the first person defines it, and is
itself defined by number relations.


[SPK] OK, but  the numbers can code noise just as they can code the
content of my 1p in this moment as I type this post. In fact it is far
more likely that it codes noise. We have to resort to all kinds of
fancy constructions to get around this fact and I find that the fact
that this must be done is a sign that something is wrong in our
thinking here.

My point is not that it is true, but that it is a consequence of the comp hyp. If you can show that 'something is wrong', then you refute comp.



 The fact that we can represent a history of events as a sequential
narrative is OK, but this is not time. Time is a measure of the change
in one aspect relative to some other that can be decided by some third
aspect. In a frozen structure there is no change, thus there is, by
definition, no time. Strings of numbers are not time just as records
of the output of a Geiger Counter is not time.

IN GR there is no time either, and even more so in most Quantum GR. At the same time you can see GR as the science of time. You are confusing God's point of view, with the relative points of view of the "terrestrial" beings.




It seems to me that from the point of view of the UD

This is ambiguous. The UD is not "really" a person. It is the
effective part of the arithmetical truth. It has no points of view.


there is no before or after or this causing that.

I have already explained that the UD defines many sort of times. The
most basic one being its own steps number, but first persons 'define'
other sort of time.


[SPK] OK, but please try to understand what I am trying to
communicate. Your definition of 'times" seems to be just a sort of
sequence, a string of numbers. How many possible strings are there?
What is the chances of an arbitrarily chosen string to code, say
Beethoven's 5th and not some randomness? See my previous claim!

Probabilities are relative to states, themselves relative to histories/ computation. Your question is meaningless. I'm afraid.





To the UD everything is simultaneously given. Additionally, the way
that the dovetailing seems to work makes it so that the UD is dense
on the space of computations in the same way that the Reals are
dense in the continuum.

Not exactly, at least for most UDs. If the Mandelbrot set is a UD,
then it is a UD dense in the space of its own version of all
computations, but it is an exceptional situation.


[SPK] Yes, but there are infinitely many such sets!

There is an infinity of UDs. But they reflect each other in a way which makes them equivalent ontologically. They have the same internal epistemologies. That is why we have to recover quantum computation from number theory.



We need a local
version of the axiom of choice that does not lead to Banach-Tarski
paradox. I think the solution is in the idea of the record keeping
that you have mentioned... The idea is that the list of properties of
a set is contained to be finite and constructable (but not necessarily
Turing computational!) so that one is not needing to assume an
infinite list of properties. Non-well founded sets allows us to do
this but that is a discussion for some other day. Peter Wegner wrote
extensively about this. http://www.cs.brown.edu/~pw/
 I am exploring this with Andrew Soltau. Hopefully we will have a
result soon.

Nice. Note that Wegner says many things "against CT", which I believe is true in the comp-physics, but irrelevant for the problem of deriving physics from numbers.



But how can this be?
  I am very interested in Eric Vandenbusche's work. I will see that
Google yields from him...

It is a young bipolar genius, of the kind "perishing (not
publishing)". His only work are notes that he wrote to me with the
solution of the first open math question in my thesis. I have put them
on my web pages. Here is the link:

http://iridia.ulb.ac.be/~marchal/Vandenbussche/AxiomatisationZ.html


[SPK] WOW! Amazing work! Please get this guy to publish in English! I
beg you!

The solution of the open problem is in the first three slides. It
shows also that G and Z are bisimulable. The other slides comes from
some questions I asked to him. It includes a pretty result showing
that the sentences asserting their own Sigma_1 truth are false (a sort
of anti-Löbian phenomenon).

[SPK] Could you elaborate on this bisimulation?

The B of the logic Z can be define in G by Bp & Dt, and the D of Z, by Bf v Dp (the D of Z is really the usual logican's notion of relative consistency). Vandenbussche found that you can dually reverse that translation: the B of G can be defined in Z by Bp v Df, and the D of G can be defined in Z by Dp & Bt. Be careful to interpret the B and D in the right logic. I should perhaps write this in the following less ambiguous (but less readable) way:

B_z A   ==   B_g A  &  D_g t
D_z A   ==   D_g A  v  B_g f

B_g A   ==   B_z A  v  D_z f
D_g A   ==   D_z A  &  B_z t

The two lines above are the usual definition of the Z box (the second follows by duality on Bp & Dt) The two last lines are Vandenbussche inversion. It leads toward an axiomatization of Z, Z1, Z* and Z1*.

So despite their very different semantics, and "hypostasic role", G and Z are variants of each other. The same for G1 and Z1, G1* and Z1*.

Unfortunately there is no such transformation available for the logics X. (X, X1, X*, X1*) We conjecture that G and X are not bisimulable, nor probably S4Grz and X.

Bruno


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



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