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: email@example.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
everything, should question causal interaction between the mental and
planes before that within the planes. The latter problems must have
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.
"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
The theorem is a special case of Stone duality, a more general
framework for dualities between topological spaces and partially
"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
in a natural way!
For example, we have
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
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
> > 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. We need the physical world to be the interface between our
separate minds, otherwise we will be trapped in the UD in endless
Poincare recursions. This is the nightmare that Nietzsche saw.
> > 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. So
far I have only seen discussions in your papers in terms of
"interviews" between different logics. What you are calling
interviews, I would call them interpretations or mappings. There is no
notion of separable entities having anything like what you and I are
doing right now here. You wrote brilliantly about your idea of
But I will continue to argue that "the logic of arithmetical self-
reference" is not an exchange of information between separate minds.
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).
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
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  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 . 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? There is
no way to define an infinitesimal or a derivative that I can find. How
do I recover the calculus? Your model has no expressions that can be
used to act as a clock... Thus it is no surprise that the whole
structure is frozen. There is no room in it for the idea of evolution,
nothing 'becomes". Everything just "is". Every fiber of my being
screams out in revulsion at this! I am not a Σ1 sentence!
> > 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
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.
> > 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!
> > 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! 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
> > 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:
[SPK] WOW! Amazing work! Please get this guy to publish in English! I
> 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?
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