Hi Roger,

The point is that there exist (provably!) statements that are infinite and thus would require proofs that can effectively inspect their infinite extent. We could argue that induction allows us to shorten the length to a finite version but this does not cover all. For instance, consider a proposed theorem that states that there exists a certain sequence of digits in the n-ary expansion of pi. How does one consider the proof of such a theorem? Constructability (by fiite means) is the key to our notions of "understanding", etc. and have lead some people to reject all math that does not admit constructable proofs. This is a HUGE problem in mathematics and by extension philosophy.



On 8/24/2012 6:39 AM, Roger Clough wrote:
Hi Stephen P. King
Hmmmm. I guess I should have know this, but if there are unproveable statements, couldn't that also mean that the axioms needed to prove them have simply been overlooked in inventorying (or constructing) the a priori ? If so, then couldn't these missing axioms be suggested by simply asking what additional axioms are needed
to prove the supposedly unproveable propositions?
Roger Clough, rclo...@verizon.net <mailto:rclo...@verizon.net>
8/24/2012
Leibniz would say, "If there's no God, we'd have to invent him so everything could function."

    ----- Receiving the following content -----
    *From:* Stephen P. King <mailto:stephe...@charter.net>
    *Receiver:* everything-list <mailto:everything-list@googlegroups.com>
    *Time:* 2012-08-23, 13:28:00
    *Subject:* Re: Emergence

    Hi Richard,

        You mean "provable statements" not "truths" per se... I guess.
    OK, I haven't given that trope much thought.... I try to keep
    Godel's theorems reserved for special occasions. It has my
    experience that they can be very easily misapplied.


    On 8/23/2012 1:24 PM, Richard Ruquist wrote:
    Stephan,

    Strong emergence follows from Godel's incompleteness because in
    any consistent system there are truths that cannot be derived
    from the axioms of the system. That is what is meant by
    incompleteness.

    Sounds like what you just said. No?
    Richard

    On Thu, Aug 23, 2012 at 1:20 PM, Stephen P. King
    <stephe...@charter.net <mailto:stephe...@charter.net>> wrote:

        Hi Richard,

            Ah! http://en.wikipedia.org/wiki/Strong_emergence

        "Strong emergence is a type of emergence in which the
        emergent property is irreducible to its individual constituents."

        OK, but "irreducibility" would have almost the same meaning
        as implying the non-existence of relations between the
        constituents and the emergent. It makes a mathematical
        description of the pair impossible... I don't think that I
        agree that it is derivable from Godel Incompleteness; I will
        be agnostic on this for now. Could you explain how it might?



        On 8/23/2012 1:10 PM, Richard Ruquist wrote:
        It is said that strong emergence comes from Godel
        incompleteness.
        Weak emergence is like your grains of sand.

        On Thu, Aug 23, 2012 at 12:48 PM, Stephen P. King
        <stephe...@charter.net <mailto:stephe...@charter.net>> wrote:

            Hi Richard,

                Pratt's theory does not address this. Could
            emergence be the result of inter-communications between
            monads and not an objective process at all? It is useful
            to think about how to solve the Sorites paradox to see
            what I mean here. A heap is said to emerge from a
            collection of grains, but is there a number or discrete
            or smooth process that generates the heap? No! The heap
            is just an abstract category that we assign. It is a name.

            On 8/23/2012 9:44 AM, Richard Ruquist wrote:

                Now if only someone could explain how emergence works.
                Can Pratt theory do that?





-- Onward!

    Stephen

    "Nature, to be commanded, must be obeyed."
    ~ Francis Bacon

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