Rich Schroeppel passed on a message from Stephen Gray:

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.

I think it is very clear that no argument of the form 1 It is impossible to know all the digits of a number other than by knowing some pattern that they follow. 2 The digits of such-and-such a number follow no pattern. 3 Therefore God, if God there be, does not know them all. 4 Therefore there is no omniscient God. can possibly be convincing to a theist, because a premise 1 is extremely doubtful, especially if it's some sort of Supreme Being that's meant to be doing the knowing; b at least in the case of numbers like sqrt(2) or e, which are the ones you seem to have in mind, premise 2 is also doubtful (e.g., because there are algorithms that will compute those digits); c even if premises 1 and 2 are accepted, the theist can simply declare that knowing all the digits is therefore a logical impossibility, in which case God's inability to do it is no more interesting than his inability to make a triangle with five sides. (To put it differently: s/he could declare that "all the digits" is not a Thing That Can Be Known.) Or, for a more concrete objection: to my mind someone knows something if they can immediately tell you it when asked; a being with access to (say) a universal Turing machine that operates outside our spacetime could do this for any question of the form "what is the Nth digit of such-and-such a number?" provided that number is computable. Whether or not "omniscient" and "God" are coherent notions, I think "intelligent being with access to resources outside our spacetime" is fairly clearly coherent -- think of Flatland -- so I really don't see how you can possibly make this sort of argument work.

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.)

It seems to me that Christians need not believe in divine omniscience if that term is defined very strongly. (Likewise for omnipotence and other omni-X qualities.) I'm sure there are some Christians with strong theological commitments to belief in omniscience -- I wouldn't be surprised if it were an official dogma of the Roman Catholic Church, for instance -- but even if you somehow succeed in demonstrating that omniscience is an incoherent notion I bet that most Christian readers' reactions will be along the lines of "hmm, OK, so apparently 'omniscient' isn't quite the right term to describe God's knowledge; fair enough".

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. There are other issues related to this, for example in what sense do numbers "exist." One might argue that they're purely human constructs, but if humans know about numbers, "God" must, also.

If numbers are purely human constructs then perhaps knowing all about them doesn't mean knowing every digit of every number that can be described; perhaps, e.g., it means knowing whatever any human being could in principle discover about them. In which case, any entity with (let's say) the ability to create, or simulate, arbitrarily many human beings who are good at mathematics would be able to "know all about" numbers. Again, it seems scarcely credible that *that* is an incoherent notion. -- g

I think that you're looking for some sort of a diagonalization argument. The way to show that there is a hierarchy of Turing-computable problems, you show that for any amount of Turing machine tape, you can simulate all Turing machines that use smaller amounts of tape & manage to disagree with all of them. You then do that again with more tape, and again with more tape, etc. The details of how storage efficient the emulation is requires a little bit of work. You can also do this with Turing machine "time" (# of steps). In a given amount of time, you can emulate (at some constant factor slowdown) any Turing machine that takes fewer steps. The details of how time efficient the emulation is requires a little bit of work. Presumably God can emulate any given mathematician, because (s)he has more time & tape & so can disagree with all of them. But we can do even better than this single hierarchy. By utilizing the concept of an "oracle", which itself can solve certain types of problems which would otherwise be insolveable, we can build an entire hierarchy of insolveable problems, and prove the existence of problems that are insolveable even with the oracle. If we iterate on this sort of thinking, we should be able to generate the sort of incoherence you are seeking. This thinking is quite similar to (and derives from) the sort of thinking that drove Cantor literally nuts. "Cantor believed his theory of transfinite numbers had been communicated to him by God." http://en.wikipedia.org/wiki/Georg_Cantor Computational complexity theory: http://en.wikipedia.org/wiki/Computational_complexity_theory Algorithmic information theory: http://en.wikipedia.org/wiki/Algorithmic_information_theory http://en.wikipedia.org/wiki/Kolmogorov_Complexity Chaitin's home page: http://www.cs.umaine.edu/~chaitin/ At 03:23 PM 12/2/2009, 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. There are other issues related to this, for example in what sense do numbers "exist." One might argue that they're purely human constructs, but if humans know about numbers, "God" must, also.

Steve Gray

Mike Stay wrote:

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On Tue, Dec 1, 2009 at 8:59 PM, <rcs@xmission.com> wrote:

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Date: Tue, 01 Dec 2009 20:49:57 -0800 Subject: Re: [math-fun] Solving polynomial equations with roots, etc. From: "Stephen B. Gray" <stevebg@roadrunner.com>

Henry Baker wrote:

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Isn't this what the Borwein expansions are all about?

At 06:07 PM 12/1/2009, you wrote:

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from SBG ... Date: Mon, 30 Nov 2009 19:44:17 -0800 Subject: Re: [math-fun] Solving polynomial equations with roots, etc. From: "Stephen B. Gray" <stevebg@roadrunner.com>

On a different subject, for anyone to comment on.

Everyone knows that sqrt(2), pi, and all irrational numbers have no decimal strings that repeat an infinite number of times. That's not necessarily the same as the decimal expansion being totally without any pattern. That is, is there any way to predict the next digits of say sqrt(2) or pi without doing one of the usual computations? In other words, is it possible to "know" the entire decimal expansion of any "ordinary" irrational? Is anything known about patterns in "regular" irrational expansions?

I'm excluding numbers invented for the sole purpose of being irrational or transcendental and with an obvious pattern like .101001000100001.... or .123456789101112..... ), etc.) I know about the question of "normal" expansions but that has little to do with my question.

Any info will be appreciated.

Steve Gray If by the Borwein expansions you mean the formula for computing hex digits without knowing the preceding ones, yes, I know about that. I also know about the contined fractions having simple patterns. Assuming that the "value" of the irrational is given only by the decimal expansion, is it possible to "know" the exact value by knowing the pattern? This is part of a theological issue which actually means very little to most people. (And it may be meaningless.). (I'm not a believer.) What do you mean by "pattern"? Do you mean an algorithm for producing each digit given only its index? If so, then any computable number works, whereas uncomputable numbers (e.g. the complexity of the rational approximations is unbounded) don't.

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