On 3/23/2012 3:44 PM, Joseph Knight wrote:
On Thu, Mar 22, 2012 at 6:40 AM, Stephen P. King
<[email protected] <mailto:[email protected]>> wrote:
On 3/21/2012 8:16 PM, Joseph Knight wrote:
On Tue, Mar 20, 2012 at 10:25 AM, Stephen P. King
<[email protected] <mailto:[email protected]>> 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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