Dear Bruno,

From: Bruno Marchal 
Sent: Monday, June 06, 2011 9:00 AM
Subject: Re: Mathematical closure of consciousness and computation
Hi Stephen, 

On 06 Jun 2011, at 05:27, Stephen Paul King wrote:

  Hi Bruno, Rex and Friends,

      My .002$...

  No theories nor machine can reach all arithmetical truth, but few  
  people doubt that closed arithmetical propositions are either true or  
  false. We do share a common intuition on the nature of arithmetical  
  I have doubt on any notion of global mathematical truth. Sets, real  
  numbers, complex numbers, etc. are simplifications of the natural  
  numbers. They are convenient fictions, I think. If we are machine, it  
  is undecidable if ontology is more than N.

      I think that there is some differences in opinion about this but it seems 
to me that we need to look at some details. For example, there should exist a 
theory that could reach all arithmetic truth given an eternity of time or an 
unnamable number of recursions or steps. 

No this cannot exist. It is precluded by the incompleteness theorem. Eternity 
can't help. Unless you take a non axiomatisable theory, or some God-like 


    Yes, you are correct. I miswrote. I had even developed an informal proof of 
this in my critique of Leibniz’ Monadology. But this still presents a 
challenge.. Umm, maybe this is where Cantor et al considered this idea in terms 
of unnamable cardinals...

  This by definition would put them forever beyond human (finite entity) 
comprehension. Whether or not there is closure or a closed form of some theory 
does not make it realistic or not. AFAIK, closed arithmetic propositions are 
tautologies, no? 
They are not tautologies, unless you mean by this "propositions true in *all* 
models of Peano Arithmetic. But then "tautology" means "theorem", and that 
would be an awkward terminology. Ax(0 ≠ s(x)) is not a tautology (it is already 
false in (Z,+), nor is Fermat last theorem.

    Yes, I did mean it that way, as in “propositions that are true in *all* 
models” but not just of Peano Arithmetic. I was considering all Arithmetics, 
especially Robinson’s. Usually one thinks of tautologies as A = A. What I am 
trying to weaken is the way that the so called law of identity is usually 
defined. I am working toward a notion of equivalence that allows for not just 
strict equality but a more general notion of “bisimilarity”. In this way 
theorems would be tautologies in this weaker form of Identity.

  That we share a common intuition of truth may follow from a common local 
measure of truth within each of us. (Here the "inside" implied by the word 
"within" is the logical/Arithmetic/abstract aspect of the duality that I 
      Additionally, we should be careful not to conflate a plurality of 
fungible individuals with a multiplicity of non-fungible entities. We can set 
up a mental hall of mirrors and generate an infinite number of self-images in 
it, but this cannot *exactly* map to all of the selves that could exist without 
additional methods to break the symmetries. 

      I have been waiting a long time for you to state this belief of yours, 
Bruno! That "Sets, real  numbers, complex numbers, etc." are simplifications of 
(mappings on/in?) the Natural Numbers. This seems to be the Pythagorean 
doctrine that I suspected that you believed. 
Would you take the time to study the papers, you would have understood that 
this is a result of comp. Comp transforms the very banal arithmetical realism 
in an authentic Pythagorean neoplatonist theology, i.e.  with some use of OCCAM 

    I am studying the papers, but I need to clarify some ideas by asking 
questions to the Professor. ;-) I do not think the way you do and must 
translate your mental language into my own to understand them.

  It has a long history and a lot of apostles that have quite spectacular 
histories. I think that there is a deep truth in this belief, but I think that 
it needs to be more closely examined.
It can be derived from Church thesis and the assumption that "we" are Turing 

    OK, but would you allow me to say that it seems that you are considering a 
form of Turing emulation that is vastly more sophisticated and subtle than the 
purely mechanical one that Turing, for example, considered with his A machines? 
The fact that you are considering infinities of computations as “running” each 
instance of us, is pushing the idea of a recursive algorithm into places it is 
never been before.

  > Perhaps there is just human belief.

  Jason said it. If you follow that slope you may as well say that there  
  is only belief by Rex. You can also decide that there is nothing to  
  explain, no theories to find, and go walking in the woods. Science, by  
  definition, assumes something beyond (human) belief.

      I admit that I laughed out loud at this! Good point, Bruno! The reduction 
of all truth to that which can be defined within a single human's belief 
trivializes and renders it meaningless. That is one of the absurd consequences 
that we lambast solipsism for, but I think that Rex should not be to swiftly 
dismissed form maybe trying to make a deeper observation; he has brought up a 
very good topic for discussion. 
      While it is absurd to reduce all truth to what a single finite entity can 
"compute" - which is that we are actually saying if we follow the 
Kleene-Turing-Church-Post road - 

Careful. It can be said that all ontological truth will be generated, but the 
epistemological truth will never be generated, but they will emerge in a not 
completely computable way. Remember that arithmetic, seen from inside, is 
*much* bigger than arithmetic see from outside.

    OK, but that poses a difficult problem because it is epistemological truth 
that we consider as reality! What we “know” to be true, even by the Bp&p 
definition, is by definition what is “real” of us individually and via 
consensus, no? I am not understanding what you mean by “arithmetic seen from 
the outside”. Are you saying that there is more to Existence than numbers? My 
apologies, I am confused.

  we are actually positing that "all truths can be defined in terms of N -> N 

That would contradict Gödel's incompleteness. Unless you mean *all* N -> N 
mappings, which is far to bigger and trivialize the theory (making it non 
testable, and unable to derive anything in physics, cognitive science, etc.).

    Not provable truths, just the ones that we can bet on. Yes, to extend to 
*all* N->N mappings would be like what we see in superstring theory – the 
landscape that has almost completely reduced SUSY to a Scholastic type of 

  Many such mappings to be sure, but N to N mappings nonetheless. We are back 
to that strange belief that Bruno explicitly, albeit inadvertently, stated. 
      But this is not really a "strange" belief, partly because it seems to be 
almost universally the default postulate within the basket of beliefs that 
people operate with in our every day world. I would like to pose the question 
of whether or not we are inadvertently painting ourselves into a corner with 
this belief. IT seems to me, and this is just a personal prejudice of mine, 
that there exists truths that cannot be named or represented exactly in terms 
of N->N maps. 
In this context, you should clearly stated if you take all N->N mappings, or 
the total computable one, or the partial computable one. The non triviality of 
comp entirely resides in such nuances.

    I do not know yet how to do that parse. I am still learning the vocabulary. 
My apologies.

  The source of this suspicion comes from what I have studied of G. Cantor's 
work on transfinites and the histrionics of practitioners of mathematical logic 
that have been examining the nature of cardinalities.
With comp, the diagonalization of Kleene gives the information. Cantor's one 
are far too crude.

    What text might you suggest that I study to understand Kleene’s 
diagonalization? I have only found this paper on the topic:

  Additionally there is my belief that the Totality of Existence must be, at 
least, Complete (not in the Gödel sense of just 1st order logics), Bicomplete 
(in the Category theory sense) and Closed (in the topological sense). This 
implies the existence of unnamable truths, or at least Truths that cannot be 
exactly represented in terms of recursive functions on the Integers.

That the totality of existence is complete seems to me to be a tautology, or a 
truth by definition.
That truth is beyond machine's means, is a theorem.
That truth is beyond us is consequence of such theorem when we assume that we 
are machines.

    Umm, not quite the same idea. I am following the reasoning in these papers: , and my 
own ideas formed from studying Category theory. The difficulty that I see in 
your definitions is that it makes the notion of a machine into something 
altogether unknowable. People seem to interpret your word machine in the same 
way that, for example, Descartes considered the idea of automata. I made that 
mistake myself until I saw that you where not considering the notion of a box 
full of levers, springs, gears and widgets.

      The question becomes one of the implications of this on our metaphysical 
assumptions about the ontologies that we are using in our thinking about the 
issue of mathematical closure of computation and consciousness. As I see it, 
and this very well could be just an eccentric thought, is that we need to be 
very careful that we do not tacitly assume that all of the minds of entities 
are replicating the same ideas as one’s own. The fact that we are continuously 
surprised at the responces that we get when we post to this List, for example, 
should be some indication that we all think differently about things and that 
when we propose the idea that consciousness is somehow some kind of N->N map or 
even some string of numbers in ℤ, then we should expect a vigorous response. 
I agree with this. There is a persisting reductionist conception of numbers and 
machines. Numbers, already just through the laws of addition and 
multiplication, transcend a lot what any consistent machines can really prove 
or even just talk about.

    Yes, but it is easy for people to not see the subtle differences that 
emerge from the stratification of levels that you are considering, as per the 8 

  BTW, did you know that ℤ *≅U(1) and  U(1) *≅ℤ  via the Pontryagin duality? 
Yes, that U(1) that is used in physics ! This is one of many reasons why I 
think that Bruno is onto something very important in his work! :-)
Well, thanks. I wish you get clearly the point someday. You might elaborate on 
the importance of U(1) ≅  Z.

    U(1) is the symmetry group that defines the electromagnetic fields in 
physics. That it is the Pontryagin dual of the integers is interesting to say 
the least. The Reals, for example, are their own Pontryagin dual. Did you 
happen to see the write up by John Baez that I linked there? This is part of 
the topological aspect that I am studying, the “concrete” side of the abstract 
ideas that you are studying. ;-)

  Do you need someone observing your brain for you to feel something?
  Why would the physical UD execution differ?
  Indeed, why would the arithmetical UD execution differ?


      Strangely enough, Bruno, in a way there is something to this idea that we 
need to consider that someone is watching for us to feel something! If we 
follow the logic of QM and accept the decoherence idea, the idea that we have a 
definite (and Boolean representable) state of the brain depends most definitely 
on the existence of what we can think of as “someone” watching: the rest of the 
With decoherence? I guess you meant with "collapse".

    There is no such a thing as “collapse” in any 3p sense. let me explain this 
idea that I am alluding to a bit more. Consider an example of it in the Quatum 
Zeno effect:

There is a nice quote there from Turing about it:

“It is easy to show using standard theory that if a system starts in an 
eigenstate of some observable, and measurements are made of that observable N 
times a second, then, even if the state is not a stationary one, the 
probability that the system will be in the same state after, say, one second, 
tends to one as N tends to infinity; that is, that continual observations will 
prevent motion …”

    The folkish form of the idea is that everything in the physical world is 
effectively observing everything else and freezing out its phase entanglement 
aspects. Nothing is truly in isolation thus the effects of decoherence 
naturally follow and generate the appearance of a classical world.

  We can break this down into a large number of mutually communicating 
observers, but that “someone is watching” has real consequences: it induces the 
2 valued definiteness that otherwise would not exist. 
You lost me.


    OK, let me loosely explain this. We have a huge number of entities that are 
sharing information about their observations of the world with each other. This 
acts to constrain the mutual “reality” that they can consider once we subtract 
our the errors and misperceptions. When we start off each observer as having a 
superposition of possible observations that they could make and then constrain 
then to have to communicate with each other such that the information in their 
communications is additive and that they have an upper bound on the time and 
computational resources available to transmit and translate those messages then 
it makes sense that what results is a bunch of binary approximations. We can 
approximate analogue signals effectively with binary – digital – 
representations of them. Nature is just pursuing its “minimize all actions” 
    Does that make some sense?

      I think that you are are reacting a bit to strongly from your Arithmetic 
Realism doctrine. 

Where do I react a bit to strongly? Perhaps it is my english? I try just to be 
short and clear. Sorry if I look like reacting strongly, or being "upset", I am 
(usually) not. Only plain dishonesty makes me nervous, but this does not seem 
to happen on this list.

    Nah, I think that your English is much better than my French! I meant that 
you are assuming that we see the ideas that you are considering with the same 
nuance and subtlety that has become natural for your thoughts.

  I would like for all of us to sit back and thought for a while exactly on 
what we are asking with this question of Mathematical closure.
I would like to insist that assuming comp, there is no mathematical closure for 
the first person points of view. The inner view of arithmetic is already beyond 
mathematics itself. Assuming comp, we might talk of theological closure (but 
this is trivial, with the greek definition of theology: the science of 
'truth'). Note that there are many argument independent of comp which could 
make us doubt of any sense for the expression of "mathematical closure". Like 
the whole of arithmetic is beyond arithmetic, the whole of math is beyond 


    Yes, I found the same lack of closure in the algebra of bisimulation that I 
have worked out (with some help). 



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