On Wed, Dec 2, 2009 at 3:23 PM, <rcs@xmission.com> wrote:
Date: Wed, 02 Dec 2009 10:55:57 -0800 Subject: Re: [math-fun] Solving polynomial equations with roots, etc. From: "Stephen B. Gray" <stevebg@roadrunner.com>
Here's the real issue. I'm trying to make an argument against the supposedly omniscient God by raising the philosophical question of whether "God," if any, instantaneously and simultaneously knows "all" the digits of Pi or sqrt(2), for example. If the digits have no pattern then God can't know all the digits because there's no such thing as "all the digits," presumably disproving omniscience. But if there were some pattern to them, a theist could argue that knowing the pattern is equivalent to knowing all the digits. If you say that this whole issue is meaningless mystical mush, I agree, but I'm trying to show by some relatively elementary mathematical- philosophical argument that a theologian could not easily dispute, that omniscience is incoherent. (I'm writing a book about Christianity.) Omniscience is usually defined by theologians as knowing all the facts that it is possible to know. It's not clear whether if God is infinite (whatever that means) he can know an infinite string of digits or even the infinite digits in "each" of the uncountable number of algebraics or transcendentals.
Assume God has access to an ideal Turing machine and has all the time he wants. Then he has a decent claim to knowledge of any computable real. In the "God knows everything that's knowable" model, God does not know the bits of the Omega number for his Turing machine. On the other hand, there's a lot of theological language used around bits of Omega. If you have some bits of Omega "revealed" to you by an "oracle", then there's no way you can prove to anyone else that they (well, more than some finite number of them) are correct. The knowledge is personal and subjective, rather like qualia. The bits of Omega are intimately related to "light" and "truth" [read "energy" and "information", via the Landauer principle that freeing up B bits of information requires at least k*T*ln 2 Joules] and "understanding" [via the principle that having a small program to enumerate a set of data points is understanding the phenomenon producing the data; as Heisenberg said in reaction to Schroedinger's claim that his wavefunction was easier to understand than matrix mechanics, "We believe we have gained intuitive understanding of a physical theory, if in all simple cases, we can grasp the experimental consequences qualitatively and see that the theory does not lead to any contradictions."]. -- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike http://reperiendi.wordpress.com