On Fri, Mar 23, 2012 at 5:12 PM, Stephen P. King <[email protected]>wrote:
> On 3/23/2012 3:44 PM, Joseph Knight wrote: > > > > On Thu, Mar 22, 2012 at 6:40 AM, Stephen P. King <[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]>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. > 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 > 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> > ). > 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. > > 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. > 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 this already. 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 assumption entails. > 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 > particulars. > 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. -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to [email protected]. To unsubscribe from this group, send email to [email protected]. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
