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.
>
> 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.
> 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
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