Hi Bruno, Rex and Friends,
My .002$...
-----Original Message-----
From: Bruno Marchal
Sent: Sunday, June 05, 2011 9:22 AM
To: [email protected]
Subject: Re: Mathematical closure of consciousness and computation
On 04 Jun 2011, at 20:03, Rex Allen wrote:
> On Sat, Jun 4, 2011 at 1:51 PM, Jason Resch <[email protected]>
> wrote:
>>
>>
>> On Sat, Jun 4, 2011 at 12:06 PM, Rex Allen
>> <[email protected]> wrote:
>>>
>>> On Sat, Jun 4, 2011 at 12:21 PM, Jason Resch
>>> <[email protected]> wrote:
>>>> One thing I thought of recently which is a good way of showing how
>>>> computation occurs due to the objective truth or falsehood of
>>>> mathematical
>>>> propositions is as follows:
>>>>
>>>> Most would agree that a statement such as "8 is composite" has an
>>>> eternal
>>>> objective truth.
>>>
>>> Assuming certain of axioms and rules of inference, sure.
>>
>> Godel showed no single axiomatic system captures all mathematical
>> truth, any
>> fixed set of axioms can at best approximate mathematical truth. If
>> mathematical truth cannot be fully captured by a set of axioms, it
>> must
>> exist outside sets of axioms altogether.
>
> Then perhaps the correct conclusion to draw is that there is no such
> thing as "mathematical truth"?
[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. 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? 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. 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.
>
> 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 - we are actually positing that "all truths can
be defined in terms of N -> N mappings". 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. 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.
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.
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.
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! :-)
>
>
>> The fractal is just an example of a simple formula leading to very
>> complex
>> output. The same is true for the UDA:
>> for i = 0 to inf:
>> for each j in set of programs:
>> execute single instruction of program j
>> add i to set of programs
>> That simple formula executes all programs.
>
> Following those instructions will let someone "execute" all
> "programs".
>
> Or, alternatively, configuring a physical system in a way that
> represents those instructions will allow someone to interpret the
> system's subsequent states as being analogous to the "execution" of
> all "programs".
[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. 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.
I think that you are are reacting a bit to strongly from your Arithmetic
Realism doctrine. 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.
Onward!
Stephen
>
>
>>> Is extraordinary complexity required for the manifestation of
>>> "mind"?
>>> If so, why?
>>>
>>
>> I don't know what lower bound of information or complexity is
>> required for
>> minds.
>
> Then why do you believe that information of complexity is required
> for minds?
If you accept that the brain is Turing emulable, then it is easy to
explain that matter/consciousness is arithmetical information as seen
from inside. That is certainly easier than explaining consciousness by
physical attributes.
>
>
>>> Is it that these recursive relations cause our experience, or are
>>> just
>>> a way of thinking about our experience?
>>>
>>> Is it:
>>>
>>> Recursive relations cause thought.
>>>
>>> OR:
>>>
>>> Recursion is just a label that we apply to some of our implicational
>>> beliefs.
>>>
>>> The latter seems more plausible to me.
>>>
>>
>> Through recursion one can implement any form of computation.
>
> But, ultimately, what is computation?
Any mathematical transformation which is Turing emulable.
Assuming Church thesis this is a very general definition.
>
>
>> Recursion is
>> common and easy to show in different mathematical formulas, while
>> showing a
>> Turing machine is more difficult. Many programs which can be
>> easily defined
>> through recursion can also be implemented without recursion, so I
>> was not
>> implying recursion is necessary for minds.
>
> Then what do you believe is necessary for minds?
A universal system, or universal numbers. Now they can be proved to
exist from logic + non negative integers, so we don't need a lot.
Jason answered "An informational process". That's OK, especially in
his context (that computation exist in math), but the word "process"
is frequently interpreted by "primitively spatio-temporal process",
when we need only (sigma_1) arithmetical relations.
Bruno
http://iridia.ulb.ac.be/~marchal/
--
You received this message because you are subscribed to the Google Groups
"Everything List" group.
To post to this group, send email to [email protected].
To unsubscribe from this group, send email to
[email protected].
For more options, visit this group at
http://groups.google.com/group/everything-list?hl=en.
--
You received this message because you are subscribed to the Google Groups
"Everything List" group.
To post to this group, send email to [email protected].
To unsubscribe from this group, send email to
[email protected].
For more options, visit this group at
http://groups.google.com/group/everything-list?hl=en.
