On 04 Jun 2011, at 19:06, Rex Allen 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.

But everyone agree on the axioms of arithmetic. And we could take any universal (in the Turing sense) system instead. The physical laws cannot depend on the choice of the "universal base". Lat us continue with (N, +, *), because it is taught in high school.




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.

Is extraordinary complexity required for the manifestation of "mind"?
If so, why?

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.


I think you are confusing computability, which is absolute (assuming Church thesis), and provability, which is always relative to theories, machines, entities, etc.

Jason is right, computation occurs in "arithmetical platonia", even in a tiny part of it actually, independently of us. This tiny part is assumed in the rest of science, and comp makes it necessarily enough (by taking seriously the first and third person distinction).

Bruno


The latter seems more plausible to me.


Rex

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