On Sat, Jun 4, 2011 at 12:06 PM, Rex Allen <rexallen31...@gmail.com> wrote:

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> 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 -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.