Hi Jason,

    Very interesting reasoning!

From: Jason Resch 
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?
**


  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!
**

  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



-- 

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

Onward!

Stephen

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