On 3/23/2012 3:44 PM, Joseph Knight wrote:


On Thu, Mar 22, 2012 at 6:40 AM, Stephen P. King <stephe...@charter.net <mailto:stephe...@charter.net>> wrote:

    On 3/21/2012 8:16 PM, Joseph Knight wrote:


    On Tue, Mar 20, 2012 at 10:25 AM, Stephen P. King
    <stephe...@charter.net <mailto:stephe...@charter.net>> wrote:


        Dear Joseph,

            How do numbers implement that necessary capacity to
        define each other and themselves? What kind of relational
        structure is necessary? From what I can tell, it looks like a
        "net of Indra" where every jewel, here a number, reflects all
        others. This is a non-well founded structure.


    You'll have to be more explicit than this if I am to make any
    sense of it.

    Dear Joseph,

        I first must say that I appreciate very much this exchange as
    it forces me to better refine my wordings and explanations. In the
    passage above I was trying to get at something that I see in the
    implied structure of numbers, given Bruno's amazing ideas.
    Remember, I "think in pictures", so the relations between numbers
    - with their Goedelizations and Loeb references - is to me a
    network where any one entity - here an integer - is defined by and
    related to all others. It looks like the structure of an infinite
    Webster Dictionary!  What I also see is that the "links" are not
    of a constant length - some connections between numbers are tiny -
    like the link between prime pairs - while others are infinitely
    long. What I am trying to point out is that this structure, is
    very much_unlike_ the structure that we think of when we just
    consider the "number line" where such a line is made up only of
    integers - 0, 1, 2, 3, ...


This is all nice, but I can't understand it unless you give make this more formal/precise.

Do you only think in words? I'm just curious... I will try harder to sketch the idea in words for you.

Think of how Goedelizations and Goedel numbers work as a visual picture, perhaps as a poitrait by Matisse or Dali. We have a string of numbers that "represents" another set of numbers *and* some arithmetic operation on those numbers. Any such Goedel number is thus the equivalent to a "handle" on the "space" of numbers (which is, by definition, a one dimensional manifold <http://en.wikipedia.org/wiki/Curve#Topology>, also see 1 <http://en.wikipedia.org/wiki/Evenly_spaced_integer_topology> and 2 <http://en.wikipedia.org/wiki/Open_sets#Topological_spaces>), therefore if it is possible to have an infinity of goedel numbers in the integers then the resulting manifold would an infinity of handles (disjoint manifolds) on it. How many unique paths would exist on such a manifold? What is the "average" length of a path? (Please recall the fact that a handle can have any size iff it is simply connected and analytic) There is no such an average for the only faithful sample of the set of possible lengths of paths is the set itself (infinite sets are isomorphic to any of their proper subsets). Remember that we can also have goedel numbers operating on (mapping into) dovetailed strings of goedel numbers and goedel numbers can have arbitrarily long number string lengths...... This makes the dimension of this manifold to be infinite because of the disjointness of the "handles" that are induced by the Goedelizing, thus making it (modulo the requirements of spaces to exist) an infinite space. It is only if the requirements of a space <http://en.wikipedia.org/wiki/Space_%28mathematics%29> not being met that this would not occur. Given that a geodelization introduces arithmetic into the set of numbers then is automatically qualifies a goedelized number line to be the dual of a space (via the Stone representation theorem <http://en.wikipedia.org/wiki/Stone_representation_theorem>).

QED.

The visual mode and the symbol mode of languages seem to have a strange conjugacy....


        Numbers as Bruno is considering them, I contend, has a
    structure that mathematicians denote as "non-well founded" in the
    sense that there is no "basic" building block out of which this
    structure is constructed unless we force it into a very tight
    straight jacket. One example of just a constraint occurs when we
    think of numbers as von Neumann numerals
    <http://bmanolov.free.fr/von-neumann-integer.php> or something
    like: s, s(), s(()), s((())), ... - where s is the null set which
    we can define in terms of Spencer-Brown's laws of form as the
    Double Cross (see
    
http://upload.wikimedia.org/wikipedia/commons/f/ff/Laws_of_Form_-_double_cross.gif),
    my point being that we only obtain a 'well-founded' version when
    we impose a constraint of the "natural' structure.


Well, of course. To talk about well-foundedness you need a class and a relation, not just a class. It doesn't make sense to say "numbers are non-wellfounded".

They are non-well founded if they are identified with Goedelizations or computations as I have shown above. I would be happy to see a counterexample, for this seems to be a proof of the non-existence of Bruno's global measure. :-(

        Before I divagate off in a ADHD haze, let me state my
    conclusion: this structure that we find as defined by COMP is very
    much unlike the nice and well behaved collection of Integers and
    should not be mistaken for it.


How? Where is the justification for this?

See my proof above.

    It is not uniform nor regular nor well-founded and I dare say that
    it is not computable in the usual sense of recursively enumerable
    mappings from N to N.


I don't even know what structure you are talking about, because you haven't defined it. If you had defined it, we could learn about it.

Its all about the Goedel numbering and the self-referencing that it induces. Loedbian operations would be even more vicious as they allow plenums (a form of continua for discrete spaces, its similar to a spectrum <http://en.wikipedia.org/wiki/Spectrum_%28functional_analysis%29>) of self-loops, which look like higher dimensional versions of totally disconnected discrete spaces, when we consider the topological view of numbers. All of this idea simply follows from the Stone duality; Logics (the 'stuff" of minds) are the ontological dual to topological Spaces (the "stuff" of physics). This is all flowing from my Prattian dualist alternative to material or ideal monist ontological theories. I see Bruno's work as being the equivalent for logic as what Newton did for physics. I mean this with all sincerity!

snip

Onward!

Stephen

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