On Sat, Jun 4, 2011 at 3:12 PM, Stephen Paul King <stephe...@charter.net>wrote:

>   Hi Jason,
>     Very interesting reasoning!

Thank you.

>   *From:* Jason Resch <jasonre...@gmail.com>
> *Sent:* Saturday, June 04, 2011 1:51 PM
> *To:* everything-list@googlegroups.com
> *Subject:* Re: Mathematical closure of consciousness and computation
> On Sat, Jun 4, 2011 at 12:06 PM, Rex Allen <rexallen31...@gmail.com>wrote:
>> On Sat, Jun 4, 2011 at 12:21 PM, Jason Resch <jasonre...@gmail.com>
>> 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.
> [SPK]
>     I see two possibilities. 1) Mathematical truth might only exist in our
> minds. But an infinity of such minds is possible...2) Might it be possible
> that our mathematical ideas are still too primitive and simplistic to define
> the kind of set that is necessary?
> **

1) More is answered by:
A: "Math -> Matter -> Minds" (or as Bruno suggests "Math -> Minds ->
Matter") than by
B: "Matter -> Minds -> Math", or
C: "Minds -> (Matter, Math)".

Compared to "B", "A" explains the unreasonable effectiveness of math in the
natural sciences, the apparent fine tuning of the universe (with the
Anthropic Principle), and with computationalism explains QM.
"C" has the least explanatory power, and we must wonder why the experience
contained within our minds seems to follow a compressible set of physical
laws, and why mathematical objects seem to posses objective properties but
by definition lack reality.

Those who say other universes do not exist are only adding baseless entities
to their theory, to define away that which is not observed.  It was what led
to theories such as the Copenhagen Interpretation, which postulated collapse
as a random selection of one possible outcome to be made real and cause the
rest to disappear.  Similarly, there are string theorists which hope to find
some mathematical reason why other possible solutions to string theory are
inconsistent, and the one corresponding to the the standard model is the
only one that exists.  Why?  They think this is necessary to make their
theory agree with observation, but when the very thing is unobservable
according to the theory it is completely unnecessary.

The situation is reminiscent of DeWitt and Everett:

> In his letter, DeWitt had claimed that he could not feel himself split, so,
> as mathematically attractive as Everett's theory was, he said, it could not
> be true. Everett replied in his letter to DeWitt that, hundreds of years
> ago, after Copernicus had made his radical assertion that the Earth revolved
> around the sun instead of the reverse, his critics had complained that they
> could not feel the Earth move, so how could it be true? Recalling Everett's
> response to him decades later, in which he pointed out how Newtonian physics
> revealed why we don't feel the Earth move, DeWitt wrote, "All I could say
> was touché!"

2) I don't know.  Godel proved that any sufficiently complex axiomatic
system can prove that there are things that are true which it cannot prove.
 Only more powerful systems can prove the things which are not provable in
those other axiomatic systems, but this creates an infinite hierarchy.
 Whether or not there is some ultimate top to it I don't know.

>> But isn't that true of nearly anything?  How many axiomatic systems are
>> there?
>> > Likewise the statement: the Nth fibbinacci number is X.
>> > Has an objective truth for any integer N no matter how large.  Let's say
>> > N=10 and X = 55.  The truth of this depends on the recursive definition
>> of
>> > the fibbinacci sequence, where future states depend on prior states, and
>> is
>> > therefore a kind if computation.  Since N may be infinitely large, then
>> in a
>> > sense this mathematical computation proceeds forever.  Likewise one
>> might
>> > say that chaitin's constant = Y has some objective mathematical truth.
>> For
>> > chaintons constant to have an objective value, the execution of all
>> programs
>> > must occur.
>> >
>> > Simple recursive relations can lead to exraordinary complexity, consider
>> the
>> > universe of the Mandelbrot set implied by the simple relation Z(n+1)=
>> Z(n)^2
>> > + C.  Other recursive formulae may result in the evolution of structures
>> > such as our universe or the computation of your mind.
> 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.
>> 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.
> [SPK]
>     Why are we sure that a “lower bound of information” or “complexity” is
> required? Seriously, there seems to be a bit of speculation from too few
> facts when it comes to consciousness!
> **

I should clarify, I don't know what the lower bound is or if there is one.

That said I do believe information and computation are importantly related
to consciousness.

> 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. 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.  For example, implementing the
> Fibonacci formula iteratively would look like:
> Fib(N)
>   X = 1
>   Y = 1
>   for int i = 2 to N:
>     i = X + Y
>     X = Y
>     Y = i
>   print Y
> This program iteratively computes successive Fibonacci numbers, and will
> output the Nth Fibbonaci number.
> Jason

There was a bug in that program, replace the last two "i"s with "j",
otherwise it breaks out of the loop too early.  :-)

>  --
> [SPK]
>     The existence of such Numbers could be a telltale sign that numbers
> require an eternal computation to define them.

I'm not sure, I can define Pi without an infinite description or
computation.  Pi = circumference of a unit circle / 2

I would agree that determining Pi from that definition probably does require
an eternal/infinite amount of computation though.


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