On Fri, Mar 23, 2012 at 5:12 PM, Stephen P. King <stephe...@charter.net>wrote:
> 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>wrote:
>> On 3/21/2012 8:16 PM, Joseph Knight wrote:
>> On Tue, Mar 20, 2012 at 10:25 AM, Stephen P. King
>>> 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
>> 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.
I do tend to think more in words or symbols than in pictures, but my intent
here is really just to get a precise understanding of what you are saying.
I hope you agree that this is necessary!
> 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
> 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
Thanks for being explicit. But as I have said before, you fail to convince
me of the relevance of these mathematical gymnastics. I can see what you
are saying, but it does not seem insightful to me.
> 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
>> 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
> They are non-well founded if they are identified with Goedelizations
> or computations as I have shown above.
It does not make sense to say things like this. A set A may be well-founded
under the relation R, but non-well-wounded under the relation S. You did
not define a relation on the set of integers, so no, you did not show that.
In any event, I cannot see how this comes remotely close to...
> 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.
Sorry, I see no justification for this claim in your proof above. Also, I
don't know what you mean when you say "this structure is defined by
COMP"...I still think you are abusing the word COMP. But I have been over
I think the opposite is true: it's a bizarre and unjustified belief to
> think that there is nothing more than particulars.
> Dear Joseph,
> Let us reason a bit about this belief. I think that it is very much
> justified simply because if one cannot name an object then statements about
> its truth or existence cannot be communicated. If a true statement about
> something cannot be communicated, is it really a truth?
I hope I don't have to list a few of the absurd beliefs that such an
> I assume, perhaps wrongly, that if an object can be named than it is, by
> definition, a "particular". Therefore, by Bp&p -> p, believing in a
> statement and that statement is true obliges me to only believe in
This clearly won't do. We can name every possible physical instance of
f=ma, or we can just name f=ma. But clearly these two situations are not
both "particulars" in the sense we are after.
> Now your comment might be restated as "it is bizarre and unjustified to
> think that existence (or there is nothing more) is nothing more than that
> which can be named".
Would you still believe the statement? I am merely trying to be consistent
> with Bruno's thesis.
I can name the statement "Abstract arithmetic truth exists".
Demonstration: Let p stand for "Abstract arithmetic truth exists".
You're just playing games with me now.
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