Hi Stephen,

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

Hi Bruno, Rex and Friends,

    My .002$...

[BM]
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
truth.
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.


[SPK]

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 entities.




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.




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 propose.) 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 razor.




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 emulable.




>
> Perhaps there is just human belief.

[BM]
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.

[SPK]

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.





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


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.).




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.




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.






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.





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.




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.





[BM]
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?

[SPK]

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 universe.

With decoherence? I guess you meant with "collapse".



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.



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.




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 mathematics.

Bruno




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



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