Frege's Concept - Script (Begriffsschrift)

[Organic]

"Frege's main goal was to improve the foundations of mathematics and scientific work in general. In attaining this goal he invented a artificial language (Begriffsschrift), which were itself a milestone in human intellectual history.

The Begriffsschrift makes use of a logical notation that makes it possible to express sentences of larger complexity than Aristotle's logic did. Indeed, we are talking a revolution in logic (it was published in 1879)."
( http://www.findlink.dk/frege/frege.htm )


Some examples of Frege's Concept-Script (Begriffsschrift) logical notations can be found here:

http://mailbox.univie.ac.at/Frank.Hartmann/Vorlesung/ws07.htm

http://www.stephenwolfram.com/publications/talks/mathml/Images/Frege.jpg

An example of 7 steps thet translates his notations to the modern linear way, can bo shown here:
http://www.roman-eisele.de/phil/stuff/logik/BaumZuBegriffsschrift.pdf

And a full text of him in modern notations, can be shown here:
http://comet.lehman.cuny.edu/mendel/papers/Adobe Versions/AdobeNewBGForms.pdf

Frege's method gives us a good opportunity to examine the general information structure that standing in the basis of well defined notations and their relations.

Most mathematicians of his time did not understand his work because he used a non-conventional way to address his remarkable ideas.

His unique representation was left behind, and the insights of it where never learned by the community of modern mathematicians.

For the last 20 years I am (on and off) in a private journey for my own pleasure that tries to find the information structures that maybe standing in the basis of any information system.

By this approach I hope to find a general method that can order these information structures by their symmetry degrees, and then to define a common and simple basis to any information system, which gives us the ability to find deeper relations between so called different areas of information systems.

Shortly speaking, I hope to find an organic and dynamic structure that can enrich the ways that self aware systems in infinity many levels of awareness, can communicate among them, without destroying or blocking each other’s opportunities to flourish.

Lately I discovered Frege’s work and I think that there is a deep connection between his representation method and my goal.

If we examine Frege’s structural forms we can find that they are private cases of broken symmetries that appearing in my ordered information structures, for example please open the attached pdf file in page 4 (in the paper, not in the acrobat screen):

http://www.geocities.com/complementarytheory/ET.pdf

Also you can see that these ordered information structures, can be represented in variety ways, but still all these different representations are based on the same symmetry transformation between its most symmetrical form and most broken form, which are limited by a given finite quantity.

The old-new way that I have started here cannot be done by a one person, so if you find my work useful to you I’ll be glad to share my best in a common research with you.

A general view of my work can be found here:

http://www.geocities.com/complementarytheory/CATpage.html


Yours,

Organic.

 

[matt grime]

So you simlutaneously want 'well defined notation' and use without explanation the following non-standard terms:

standing in the basis of
information system
symmetry degree
ordered information structure

You also cite the work of a man whose contributions were overlooked (allegedly) because he didn't use conventional notation. You might want to ponder on that the next time you write binary tree for something that isn't a tree (in the mathematical sense).

Given your repeated indications that you think mathematicians are 'wrong' and closed minded about your ideas, who do you think is going to come and help you?

 

[Organic]

I can find persons that can understand my ideas, and these persons do not afraid to open themselves and their methods to the complexity of the real life, for example:

-----Original Message-----
From: Dr. A.M.Selvam [mailto:amselvam@eth.net]
Sent: Monday, March 01, 2004 10:04 AM
To: Shadmy Doron
Subject: A new approach for the definition of a NUMBER

1 March 2004

Dear Doron  Shadmi 

 I am indebted to you for your email dated 
16 February giving references of your valuable 
research work.
 
 I find your original research work very valuable 
for developing a simple unified theory with ramifications
in the numerical modeling of nonlinear dynamical 
systems/processes. 

 Your research work would benefit many of the scientists 
particularly those who are working in the area of 
numerical modeling.

                                   with best regards
                                    yours sincerely
                              Dr. (Mrs.) A. Mary Selvam

Papers of Dr. (Mrs.) A. Mary Selvam can be found here:

http://www.geocities.com/CapeCanaveral/Lab/5833/pub11.html

 

[Organic]

Matt,

Let us try to work together.

symmetry degree
ordered information structure

Let F be a finite integer which its tree-like structure can be ordered by its symmetry degree, where symmetry degree means transformation between complete parallel branches to complete nested branches.

Please take these informal definitions and try to address them in a rigorous way.

Thank you.

Organic

 

[matt grime]

Why would I be remotely qualified to offer a rigorous definition of the objects in your head? You aren't Witten, sorry to have to break that to you, we don't have to decide what you mean. You do. You've also, as ever, just introduced yet more undefined terms into it. Remember you are the one that wanted 'well defined notation'.

I've read some of her papers. Not knowing what 'quantum chaoslike' means (despite knowing what quantum chaos is) I don't feel qualified to offer an opinion on her merits. Presumably you checked all her papers carefully, and understood them all. Also her statements about occurences of things in DNA patterns has to be up there in the 'stating the bleeding obvious' category, but then she is a meteorologist.

 

[Organic]

Matt,

All I asked is to work together with you, but since you don't find my ideas understood by you, then let's forget my offer.

I have noticed that you never opened a thread of your own, can you tell us why?

but then she is a meteorologist.

I think that's the reason why she can understand me and you don't, because your community trying to avoid reality influence.

 

[matt grime]

Offer me a simple clear definition of any of the things you use and we can see.

Why would I bother starting a thread in TD? I've no need to, or interest in so doing. My research is available to those who need it. Somehow I don't think asking in General Math whether the homotopy colimit in the stable category is naturally isomorphic to the first corner in the triangle of the Bousfield localization wrt the f.g modules will get any responses worth considering. If i come across anything that requires posting I'll be the first to ask.

 

[Organic]

Can you take this transformation and address in your way?

[b]
A set is only a framework to explore our ideas.

The concept of an oredered set does not depend on the quantity concept as shown here:

By Complementary Logic multiplication is noncommutative,
but another interesting result is the fact that multiplication 
and addition are complementary opreations that can be ordered 
by different symmetry degrees where quantity remains unchanged 
for example:

A Number is anything that exists in ({},{__})

Or in more formal definition:

({},{_}):={x|{} <-- x(={.}) AND x(={._.})--> {_}}

Where -->(or <--) is ASPIRATING(= approaching, but cannot become closer to).

If x=4 then number 4 example is:

Number 4 is a fading transition between multiplication 1*4 and 
addition ((((+1)+1)+1)+1) ,and vice versa. 

This fading can be represented as:
 

(1*4)              ={1,1,1,1} <------------- Maximum symmetry-degree, 
((1*2)+1*2)        ={{1,1},1,1}              Minimum information's 
(((+1)+1)+1*2)     ={{{1},1},1,1}            clarity-degree
((1*2)+(1*2))      ={{1,1},{1,1}}            (no uniqueness) 
(((+1)+1)+(1*2))   ={{{1},1},{1,1}}
(((+1)+1)+((+1)+1))={{{1},1},{{1},1}}
((1*3)+1)          ={{1,1,1},1}
(((1*2)+1)+1)      ={{{1,1},1},1}
((((+1)+1)+1)+1)   ={{{{1},1},1},1} <------ Minimum symmetry-degree,
                                            Maximum information's  
                                            clarity-degree                                            
                                            (uniqueness)


============>>>

                Uncertainty
  <-Redundancy->^
    3  3  3  3  |          3  3             3  3
    2  2  2  2  |          2  2             2  2
    1  1  1  1  |    1  1  1  1             1  1       1  1  1  1
   {0, 0, 0, 0} V   {0, 0, 0, 0}     {0, 1, 0, 0}     {0, 0, 0, 0}
    .  .  .  .       .  .  .  .       .  .  .  .       .  .  .  .
    |  |  |  |       |  |  |  |       |  |  |  |       |  |  |  |
    |  |  |  |       |__|_ |  |       |__|  |  |       |__|_ |__|_
    |  |  |  |       |     |  |       |     |  |       |     |
    |  |  |  |       |     |  |       |     |  |       |     |
    |  |  |  |       |     |  |       |     |  |       |     |
    |__|__|__|_      |_____|__|_      |_____|__|_      |_____|____
    |                |                |                |
    (1*4)            ((1*2)+1*2)      (((+1)+1)+1*2)   ((1*2)+(1*2))
 
 4 =                                  2  2  2
          1  1                        1  1  1          1  1
   {0, 1, 0, 0}     {0, 1, 0, 1}     {0, 0, 0, 3}     {0, 0, 2, 3}
    .  .  .  .       .  .  .  .       .  .  .  .       .  .  .  .
    |  |  |  |       |  |  |  |       |  |  |  |       |  |  |  |
    |__|  |__|_      |__|  |__|       |  |  |  |       |__|_ |  |
    |     |          |     |          |  |  |  |       |     |  |
    |     |          |     |          |__|__|_ |       |_____|  |
    |     |          |     |          |        |       |        |
    |_____|____      |_____|____      |________|       |________|
    |                |                |                |
(((+1)+1)+(1*2)) (((+1)+1)+((+1)+1))  ((1*3)+1)        (((1*2)+1)+1)

   {0, 1, 2, 3}
    .  .  .  .
    |  |  |  |
    |__|  |  |
    |     |  | <--(Standard Math language uses only this
    |_____|  |     [i]no-redundancy_no-uncertainty_symmetry[/i])
    |        |
    |________|
    |    
    ((((+1)+1)+1)+1)
 

Multiplication can be operated only among objects with structural identity, 
where addition can be operated among identical and non-identical 
(by structure) objects.

Also multiplication is noncommutative, for example:

2*3 = ( (1,1),(1,1),(1,1) ) , ( ((1),1),((1),1),((1),1) )

3*2 = ( (1,1,1),(1,1,1) ) , ( ((1,1),1),((1,1),1) ) , ( (((1),1),1),(((1),1),1) )
[/b]

 

[matt grime]

No, because I shouldn't have to tell you what the definitions are of the objects in your own theory, and more importantly I think calling it a multiplication on N is a misleading thing to do. Even allowing for your defintion of elements of n, the product of two of them is not another element of N. What is 2*3, because it's not an integer. As lots of people have pointed out you are defining some not uninteresting combinatorial constructions on certain kinds of trees, probably making it something akin to a groupoid, or monoid. Some elements get labels in N, some do not.

A reasonable mutliplication takes pairs of elements in N and assigns another element in N (closure), we might also ask that it distributes over addition, and that m*n = m*(n-1) +m or that m*1=1*m=m

We don't have to do that, but all the structures you are describing are known and studied (rings, division algebras, algebras, domains, fields, rigs rngs, monads, groups, groupoids).

In fact the reason why we identify copies of the integers inside these structure is exactly because given a multiplicative identity where mulitplication distributes over addition (ie a ring structure) then we either get a copy of Z in the ring, ro we get some ring of integers modulo M for some M.

If we don't require associativity etc then we get other well known objects.

 

[Organic]

We don't have to do that, but all the structures you are describing are known and studied (rings,
division algebras, algebras, domains, fields, rigs rngs, monads, groups, groupoids).

I take these structures as general representation of information, which is ordered by its symmetry levels that are
connected to information's clarity degrees, based on uncertainty AND redundancy.

Can you show me some mathematical research that using multiplication and addition from this point of view?

Shortly speaking, can you show me this?:

(1*4)              ={1,1,1,1} <------------- Maximum symmetry-degree, 
((1*2)+1*2)        ={{1,1},1,1}              Minimum information's 
(((+1)+1)+1*2)     ={{{1},1},1,1}            clarity-degree
((1*2)+(1*2))      ={{1,1},{1,1}}            (no uniqueness) 
(((+1)+1)+(1*2))   ={{{1},1},{1,1}}
(((+1)+1)+((+1)+1))={{{1},1},{{1},1}}
((1*3)+1)          ={{1,1,1},1}
(((1*2)+1)+1)      ={{{1,1},1},1}
((((+1)+1)+1)+1)   ={{{{1},1},1},1} <------ Minimum symmetry-degree,
                                            Maximum information's  
                                            clarity-degree                                            
                                            (uniqueness)

Let us say that we have here a transformation between super set {1,1,1,1} to "normal" set {{{{1},1},1},1}
and vise versa.

 

[matt grime]

Seeing as you haven't told anyone what 'clarity degree' 'symmetry degree' 'redundancy' 'uncertainty' mean to you, then the answer is trivially 'of course I can't show you any of that, because no one knows what to show you'.

Show me a snark, or a boodjum, something gambolling in the mimsy wabe.

Try looking up, oh, I don't know, theory of non-associative algebras, monads or something.

 

[Organic]

you haven't told anyone what 'clarity degree' 'symmetry degree' 'redundancy' 'uncertainty' mean to you,

This is what it means to me:

              Uncertainty
  <-Redundancy->^
    3  3  3  3  |
    2  2  2  2  |
    1  1  1  1  |
   {0, 0, 0, 0} V
    .  .  .  .
    |  |  |  |
    |  |  |  |
    |  |  |  |
    |  |  |  |
    |  |  |  |
    |__|__|__|_
    |
    (1*4)


   {0, 1, 2, 3}
    .  .  .  .
    |  |  |  |
    |__|  |  |
    |     |  | <--(Standard Math language uses only this
    |_____|  |     no-redundancy_no-uncertainty_symmetry)
    |        |
    |________|
    |
    ((((+1)+1)+1)+1)

Is it understood?

 

[matt grime]

No. That is an example in one particular case.

Is redundancy the number of columns, the labels on the columns, the shape of the columns (same for uncertainty as in all that follows)

it doesn't say why there is no redundancy in the second diagram.

it doesn't tell me what the redundancy of another diagram might be.

it doesn't tell me if redundancy is something only attributable to trees, or if other objects may have it, whatever 'it' is.

i don't know how to 'measure' redundancy, is ti a numerical qunatity like the order of an element of a group.

so tell us what constitutes redundancy.

 

[matt grime]

Here's an example. Pick some meaningful word. Solid, and hence solidity. They sound as though they may have vaguely mathematical meanings.

Let me now associate to them some proper definition of my own that I will refuse to explain. Ok, done it. R is solid, Q is not solid. Question: is N solid?

 

[Organic]

Let us get back to set and super set:

Let us say that we have here a transformation between super set {x,x,x,x} to "normal" set {{{{x},x},x},x}
and vise versa.

Let a,b,c,d stends for uniquness, then we get:

              Uncertainty
  <-Redundancy->^
    d  d  d  d  |
    c  c  c  c  |
    b  b  b  b  |
   {a, a, a, a} V
    .  .  .  .
    |  |  |  |
    |  |  |  |
    |  |  |  |
    |  |  |  |
    |  |  |  |
    |__|__|__|_
    |
    ={x,x,x,x}


   {a, b, c, d}
    .  .  .  .
    |  |  |  |
    |__|  |  |
    |     |  | <--(Standard Math language uses only this
    |_____|  |     no-redundancy_no-uncertainty_symmetry)
    |        |
    |________|
    |
    ={{{{x},x},x},x}

[b]
============>>>

                Uncertainty
  <-Redundancy->^
    d  d  d  d  |          d  d             d  d
    c  c  c  c  |          c  c             c  c
    b  b  b  b  |    b  b  b  b             b  b       b  b  b  b
   {a, a, a, a} V   {a, a, a, a}     {a, b, a, a}     {a, a, a, a}
    .  .  .  .       .  .  .  .       .  .  .  .       .  .  .  .
    |  |  |  |       |  |  |  |       |  |  |  |       |  |  |  |
    |  |  |  |       |__|_ |  |       |__|  |  |       |__|_ |__|_
    |  |  |  |       |     |  |       |     |  |       |     |
    |  |  |  |       |     |  |       |     |  |       |     |
    |  |  |  |       |     |  |       |     |  |       |     |
    |__|__|__|_      |_____|__|_      |_____|__|_      |_____|____
    |                |                |                |
    {x,x,x,x}        {x,x},x,x}       {{{x},x},x,x}    {{x,x},{x,x}}     
 
                                      c  c  c
          b  b                        b  b  b          b  b
   {a, b, a, a}     {a, b, a, b}     {a, a, a, d}     {a, a, c, d}
    .  .  .  .       .  .  .  .       .  .  .  .       .  .  .  .
    |  |  |  |       |  |  |  |       |  |  |  |       |  |  |  |
    |__|  |__|_      |__|  |__|       |  |  |  |       |__|_ |  |
    |     |          |     |          |  |  |  |       |     |  |
    |     |          |     |          |__|__|_ |       |_____|  |
    |     |          |     |          |        |       |        |
    |_____|____      |_____|____      |________|       |________|
    |                |                |                |
    {{{x},x},{x,x}} {{{x},x},{{x},x}} {{x,x,x},x}      {{{x,x},x},x} 

    a, b, c, d}
    .  .  .  .
    |  |  |  |
    |__|  |  |
    |     |  | <--(Standard Math language uses only this
    |_____|  |     no-redundancy_no-uncertainty_symmetry)
    |        |
    |________|
    |    
    {{{{x},x},x},x}
[/b]

 

[matt grime]

Still doesn't explain what it is and how to find it. Firstly I doubt if ever learned the rules that allow you to assign the letters abc and d to those patterns. Secondly the last picture has no redundancy or uncertainty in it, an the first one does. but what about the others? All you need to say is that the diagram has redundancy if ... blah, where that's some description, perhaps only for the n=4 case, youi can generalize later.

Is it perhaps that the diagram is 'certain' iff there is one row of letters when you label it? If so why didn't you just say so. Can you give an equivalent statement that telss me when a diagram as redundancy.

Note I don't need to know what redundancy *means*, only the criteria for noting its existence.

For instance one doesn't need to understand the meaning of the *word* chaos to know that a dynamical system is chaotic if it has topological transitivity and sensitive dependence on initial conditions, all of which are well defined.


Have you deicded if N is solid yet?

 

[Organic]

Matt,

There is an algorithm for this, which is based on Cartesian product.

You can see it here:

http://cyborg2000.xpert.com/ctheory/

Please don't go beyond 6 or 7.

The cartesian pruduct results adding left-right combinations that can be ignored.

 

[matt grime]

So are you actually going to say what you mean by redundancy and uncertainty or not? Come on, Organic, it's your requirement to have things well defined.

Figured out if N is solid or not

 

[Organic]

Let us say that we have here a transformation between
super set {x,x,x,x} to "normal" set {{{{x},x},x},x} and vise versa.


Let XOR be #

Let a,b,c,d stends for uniquness, then we get:

              Uncertainty
  <-Redundancy->^
    d  d  d  d  |
    #  #  #  #  |
    c  c  c  c  |
    #  #  #  #  |
    b  b  b  b  |
    #  #  #  #  |
   {a, a, a, a} V
    .  .  .  .
    |  |  |  |
    |  |  |  |
    |  |  |  |
    |  |  |  |
    |  |  |  |
    |__|__|__|_
    |
    ={x,x,x,x}


   {a, b, c, d}
    .  .  .  .
    |  |  |  |
    |__|  |  |
    |     |  | <--(Standard Math language uses only this
    |_____|  |     no-redundancy_no-uncertainty_symmetry)
    |        |
    |________|
    |
    ={{{{x},x},x},x}

[b]
============>>>

                Uncertainty
  <-Redundancy->^
    d  d  d  d  |          d  d             d  d
    #  #  #  #  |          #  #             #  #        
    c  c  c  c  |          c  c             c  c
    #  #  #  #  |          #  #             #  #   
    b  b  b  b  |    b  b  b  b             b  b       b  b  b  b
    #  #  #  #  |    #  #  #  #             #  #       #  #  #  #   
   {a, a, a, a} V   {a, a, a, a}     {a, b, a, a}     {a, a, a, a}
    .  .  .  .       .  .  .  .       .  .  .  .       .  .  .  .
    |  |  |  |       |  |  |  |       |  |  |  |       |  |  |  |
    |  |  |  |       |__|_ |  |       |__|  |  |       |__|_ |__|_
    |  |  |  |       |     |  |       |     |  |       |     |
    |  |  |  |       |     |  |       |     |  |       |     |
    |  |  |  |       |     |  |       |     |  |       |     |
    |__|__|__|_      |_____|__|_      |_____|__|_      |_____|____
    |                |                |                |
    {x,x,x,x}        {x,x},x,x}       {{{x},x},x,x}    {{x,x},{x,x}}     
 
                                      c  c  c
                                      #  #  #      
          b  b                        b  b  b          b  b
          #  #                        #  #  #          #  #         
   {a, b, a, a}     {a, b, a, b}     {a, a, a, d}     {a, a, c, d}
    .  .  .  .       .  .  .  .       .  .  .  .       .  .  .  .
    |  |  |  |       |  |  |  |       |  |  |  |       |  |  |  |
    |__|  |__|_      |__|  |__|       |  |  |  |       |__|_ |  |
    |     |          |     |          |  |  |  |       |     |  |
    |     |          |     |          |__|__|_ |       |_____|  |
    |     |          |     |          |        |       |        |
    |_____|____      |_____|____      |________|       |________|
    |                |                |                |
    {{{x},x},{x,x}} {{{x},x},{{x},x}} {{x,x,x},x}      {{{x,x},x},x} 

    a, b, c, d}
    .  .  .  .
    |  |  |  |
    |__|  |  |
    |     |  | <--(Standard Math language uses only this
    |_____|  |     no-redundancy_no-uncertainty_symmetry)
    |        |
    |________|
    |    
    {{{{x},x},x},x}
[/b]

For clearer picture please read this:
http://www.geocities.com/complementarytheory/HelpIsNeeded.pdf

Figured out if N is solid or not

No input has solid form.

 

[matt grime]

Nope, you keep drawing the same pictures, but you still won't tell us about redundancy and uncertainty.How can you state no input has solid form. I haven't told you what the definition of 'solid' is. You're presuming that it is something to do with the ordinary meaning of the word solid. It isn't really. Here's a hint:

N, Z, R, C are 'solid', Q is not, neither is the set {1/n | n in N}, the cantor set is 'solid' surprisingly.

So is the set of finite groups solid?

 

[Organic]

Matt,

Please give your definition to redundancy and uncertainty as you understand from my pictures.

For me solid is what I call the full set {__} which is a one and only one pointless element.


For more information about redundancy and uncertainty please look at:

http://www.geocities.com/complementarytheory/CATheory.pdf

from page 7 (in the paper, not in the acrobat screen).

 

[matt grime]

Don't you get it that I don't have a definition of any of those terms from your diagram? I cannot infer one. Are you about to accuse me of being stupid for this? I'd like to remind you that in our little parallel you've not managed to predict what my definition of 'solid' means.

You draw one picture and just state that the columns are the uncertainty, and the rows are the redundancy or vice versa. Then you draw another diagram that has the same number of columns and a different number of rows, using some algorithm and there the rows and columns do not correspond to redundancy or uncertainty. Just tell me what these terms are!

Here:

|__|<filler space>|__|
|_____________|
|
|

what is there any uncertainty or redundancy in that diagram?

Page 7 refers to the 'uncertainty' concept. Which is what?

And there's a mistake there where you refer to the cardinality of R as being strictly less than c, when by definition it is c.

I have some tree. What is the redundancy of that tree? What is the uncertainty of that tree. Is it some function on the nodes, the leaves, the branches? What is it measeured in, if it's measured. Somethings aren't measured on scales, obviously, so what property must it demonstrate for us to say it is uncertain? For instance, the logistic map kx(x-1) displays chaotic behaviour for certain k, in precisely defined terms of its topological properties, so perhaps I need to ask what is involved in uncertainty and redundancy. What objects have these properties, whatever they may be. Tell us, we want to know what you mean.

 

[Organic]

You did not understand what I wrote.

Please read http://www.geocities.com/complementarytheory/CATheory.pdf
form page 7 (Introduction) until the and of the article.

 

[matt grime]

I have a crappy dial up connection I don't want to download that again. If you are so uninterested in explaining your ideas here, then perhaps I can't be bothered to go fetch that paper either.

as it is i have a copy lying around.

RU (redundancy uncertainty) is the first CR in the AL

how can it be two distinct things and take the the first person singular?

CR is the computational root it is EP in AL

EP is the explorable product

AL is the assiciation levelSo, what is the association level in a tree, and what is the explorable product? You state when it exists but you don't state what it is.

So I asked you to explain it here, demonstrated I've read your article, so how about meeting us halfway and explaining it here to save me having to read that pdf again.Still not got an answer as to what solid means?

I'll tell you:

A metric space is solid iff it is complete ie all cauchy sequences converge (that is not your definition of complete by the way, and has nothing to do with your opinions). N, R and C are all solid, Q is not (with the obvious metrics)

 

[Organic]

Dear Matt,

A metric space is solid iff it is complete ie all cauchy sequences converge (that is not your definition of complete by the way, and has nothing to do with your opinions). N, R and C are all solid, Q is not (with the obvious metrics)

Again for me solid is what I call the full set {__} which is a one and only one pointless/segmentless element.

We cannot find any form of finite or infinitely many elements in it so all Cauchy sequences cannot cover the full set.

My number system elements are based on association between segments and points
( {._.} AND {.} ) that first of all can be identified by their structural forms.

Please look at this table:
http://www.geocities.com/complementarytheory/ETtable.pdf

If you don't have any interest to develop this approach, then it is ok with me.

There is no halfway here, because it is a paradigm's change in Math language.

 

[Organic]

Introduction


If we examine the content of a set in terms of the symmetry concept, we can find at least two levels of symmetries that can be ordered by their simplicity degrees.

The most symmetrical and simplest content is Emptiness, which is represented by the empty set notation { } = content does not exist.

On top of this simplicity we can define two opposite types of symmetry contents,
{__} and {._. .} .

Let power 0 be the simplest level of existence of some set's content.

{__} is an infinite non-localized element (which is a one and only one pointless/segmentless element) that notated as 0^0 = 1 (1 continuum)

(A segment is the shortest interval that existing between any given points)

{._. .} is infinitely many elements that are notated as oo^0 = 1 (connector XOR point)

(connector is any segment which existing between any two given points)

So, what we get is this basic information structure:  {._. .}   {___}
                                                             { }

Let { } be E simplicity or Esim (E for Emptiness).

Let {___} be Csim (C for Continuum).

Let {._. .} be Dsim (D for Discreteness).

~ = NOT

Any transformation from {} to {__} or {._. .} is based on phase transition, because we have |{}|(=0) to ~|{}|(=~0) transition.

A Csim and a Dsim are opposites because Csim is a one continuum and Dsim is finite or infinitely many elements.

The above identification is based on the structure property, and it can not be done by the quantity property, because Csim XOR Dsim are exclusively = 1 .

So, in the case of the symmetry concept, the structure property is more sensitive than the quantity property, when we examine them by the information concept.

If we want to go beyond the information about the existence of Csim XOR Dsim,
we have to associate between them, by changing XOR to AND connective.

By doing this we can define elements that have properties, which are combinations of Esim, Csim and Dsim.



Let us find a definition for existence under Complementary Associations Theory (CAT) :

Existence

Definition AA: Un-explorable Existence is a state of some opposite concepts,
before there is any mutual influence on each opposite's property.

Example: Light before turning into darkness, darkness before turning into light.


Definition BB: Explorable Existence is a state of some opposite concepts,
where there is a mutual influence between their opposite properties.

Example: Light turning into darkness, darkness turning into light.



Let us write the CAT's axiom and definitions.

The Axiom of exploration:
Explorable is any association between Csim AND Dsim.

Definition A:
Explorable Product (EP) exists iff it is an association between Continuum (Csim) and Discreteness (Dsim) concepts, so CD is Csim AND Dsim .


Now, let us answer some questions:

Q: What is an "association"?
A: Association is any possible mutual influence between opposite concepts.

Q: What is an "explorable product"?
A: According to definition BB, it is the element coming from association,
and it can be explored.

Definition B:
Association-Level (AL) is an invariant quantity, being kept through CD associations.

Definition C:
Computational Root (CR) is EP in AL.

(An example of definitions B and C:
 .   .   .       .   .   .       .   .   .
 |   |   |       |   |   |       |   |   |
 |   |   |       |___|_  |       |___|   |
 |   |   |       |       |       |       |
 |___|___|_      |_______|       |_______|
 |               |               |
 CR quantity is being kept through CD associations)

Definition D:
Redundancy and Uncertainty (RU) concepts, are used as invariant structural degree of CR, determining its exact position in AL (there is an algorithm for this).


Definition E:
Full RU (FRU) is the first CR in AL.

Definition F:
Not RU (~RU) is the last CR in AL.

Definition G:
Partial RU (PRU) is any CR which is not FRU and not ~RU.

An example of definitions E, F and G:
 .   .   .       .   .   .       .   .   .
 |   |   |       |   |   |       |   |   |
 |   |   |       |___|_  |       |___|   |
 |   |   |       |       |       |       |
 |___|___|_      |_______|       |_______|
 |               |               |
 FRU CR             PRU CR             ~RU CR

A general graphic description of a CR

     .     .     .<------ D (Discreteness)
     |     |     |
     |     |     |
     |     |     |<------ The association between CD
     |     |     |
     |     |     |
     |_____|_____|__<---- RU marker
     |  ^
     |   \____ C (Continuum)
     |
     |<---- Next-AL marker

If we connect ideas coming from Information Theory and Topology, then we can use a concept like symmetry, to describe a connection between structure and information's clarity-degree, for example:

   <-Redundancy->
    c   c   c  ^<----Uncertainty
    b   b   b  |    b   b
    a   a   a  |    a   a   c       a   b   c
    .   .   .  v    .   .   .       .   .   .
    |   |   |       |   |   |       |   |   |
    |   |   |       |___|_  |       |___|   |
    |   |   |       |       |       |       |
    |___|___|_      |_______|       |_______|
    |               |               |
       FRU CR             PRU CR             ~RU CR

The uncertainty is based on XOR connective between a,b,c,...

This example describes AL 3 and we can see the connection between structure's symmetry-degree and information's clarity-degree, determining CR's exact position in AL 3.

Because RU concepts are not used in ZF or Peano's axioms, all their number system is limited to ~RU CR, therefore their number system is a proper subset of CAT's number system.

Some claim that using sequences instead of sets can do it, but we use the RU concepts as the fundamentals of all information structures, where ~RU is a private case of some information structure, that used as the information structure of ZF or Peano's axioms.

 

[matt grime]

Originally posted by Organic
Dear Matt,

Again for me solid is what I call the full set {__} which is a one and only one pointless/segmentless element.

We cannot find any form of finite or infinitely many elements in it so all Cauchy sequences cannot cover the full set.

My number system elements are based on association between segments and points
( {._.} AND {.} ) that first of all can be identified by their structural forms.

If you don't have any interest to develop this approach, then it is ok with me.

There is no halfway here, because it is a paradigm's change in Math language.

I don't think you quite got the point of this exercise. Whatever you think the word 'solid' might mean doesn't matter here, because I gave it a definition, in atttempting to tell you what the definition is I only gave you isolated examples, just like you do, and refused to offer the definition, again just like you. I could equally have called the property 'wibbliness' for the purposes of the exercise. I only chose to use 'solid' because of its deliberately confusing nature by being a meaningful word already with pseudo-mathematicl implications.

As I was doing this mathematically your ignorance of Cauchy Sequences and complete metric spaces does not matter: the Real numbers are rigorously constructed by taking the completion of Cauchy sequences of rationals (or Dedekind cuts).

So go back and replace the word solid by wibbly if you like, or orange. I was attempting to show that just coining a phrase and giving select examples of things to do with that phrase does not offer a proper definition, and this time part of your precis is that you want well defined notation.

I am not interested at the moment in what the property of uncertainty or whatever means for a set's properties, just in attempting to find a way to decide if some set/tree possesses a degree of uncertainty. That requires some philosphical thought. I am attempting to convey to you that to begin with it isn't important for me to know all that the property implies, just what implies the property. Suppose I were to ask what a group is. I don't need a big talk about symmetries and orders and so on, I just need to know it is a ste with a binary operation satisfying 4 axioms - I can then identify what things are groups. Later I can begin to worry about what it means to be a group.






You gave me an analogy of taking a cat apart once. You think putting a cat before me and saying that is a cat is defining a cat - it only gives an example of a cat. Suppose I put in front of you a canivorous quadruped, such as a dog. How is that not a cat too if all I have to go on is the single example you offered to me? You can't say it looks different, because I might produce a different coloured cat and that looks different too, or a tiger, if you showed me a domestic cat.

It is a bad analogy anyway as taxonomy is not an exact science - there are fuzzy boundaries.

 

[Organic]

Dear Matt,

Math language is also not an exact science but only a rigorous agreement between a group of people that can be chaneged when paradigm is chaneged.

For example N members are only structural private case of:

http://www.geocities.com/complementarytheory/ETtable.pdf

Some examples of Frege's Concept-Script (Begriffsschrift) logical notations can be found here:

http://mailbox.univie.ac.at/Frank.Hartmann/Vorlesung/ws07.htm

http://www.stephenwolfram.com/publications/talks/mathml/Images/Frege.jpg

An example of 7 steps thet translates his notations to the modern linear way, can bo shown here:
http://www.roman-eisele.de/phil/stuff/logik/BaumZuBegriffsschrift.pdf

So as you see the structure of current logical system can be richer then it is.

 

[matt grime]

Given your repetition of this view, how do reconcile that with the fact you were prepared to tell me what I meant by 'solid' or 'wibble' when you didn't know what the convention I had adopted meant? If your private language is just that and no one knows what you mean then you will keep on failing to get many people interested. Remember it is you who asked for well defined notation. So define everything. You've already seen the problems you get into if you presume to redefine the terms in cantor's argument. There are definitions. If you want to use the same name for something else then you must say so, and not expect the old statements to remain true. Why on Earth it is a problem that when you redefine every term in a mathematical argument you generate contradictions is still a mystery.

 

[Organic]

Matt,

For me it is very simple.

No "exact" definition in the world can construct a solid line by using infinitely many elements, exactly as no infinitely many pieces of glass can construct a solid glass.

There is a qualitative difference between the two states that no "quantitative oriented" definition can define.

You wrote:

sin(1/x) is not trivially defined at zero unless we declare its value to be something specific; no assignment of a value makes it continuous at zero. The real valued function sqrt(x) is not defined for x negative, there are no sinuglarities in sight.

Please explain what do you mean by "sqrt(x) has no singularities in sight."?

Or what do you mean by "sin(1/x) is not trivially defined at zero unless we declare its value to be something specific"?

 

[Organic]

p and q are real numbers.

If p < q then [p, q] = {x : p <= x <= q} or
(p, q] = {x : p < x <= q} or
[p, q) = {x : p <= x < q} or
(p, q) = {x : p < x < q} .

A single-simultaneous-connection is any single real number included in p, q
( = D = Discreteness = a localized element = {.} ).

Double-simultaneous-connection is a connection between any two real numbers included in p, q ( = C = Continuum = a non-localized element = {.___.} ).

Therefore, x is . XOR .___.

In Conventional Math 0^0 is not well defined, because each member is D.

Let us say that power 0 is the simplest level of existence of some set's content.

Because there are no Ds in C, its base value = 0, but because it exists (unlike the emptiness), its cardinality = 0^0 = 1.

There are now 3 kinds of cardinality:

|{}| = 0 = the cardinality of the Empty set.

|{._.}| = 0^0 = 1 = the cardinality of C.

|{.}| = 1^0 = 1 = the cardinality of D.

Any point is a D element. Any line is a C element.

It means that there is a XOR connective between LINES and POINTS.

XOR connective between LINES and POINTS
0(LINE) 0(POINT) -> 0-(No information) -> no conclusion.
0(LINE) 1(POINT) -> 1-(Clear Particle-like information) -> conclusions on points.
1(LINE) 0(POINT) -> 1-(Clear Wave-like information) -> conclusions on lines.
1(LINE) 1(POINT) -> 0-(No clear information) -> no conclusion.

We can break C infinitely many times, but always we shall find an invariant structural state of {._.}, which is a connector between any two Ds.
Let power 0 be the simplest level of existence of some set's content.

{._. .} = Dinf = Infinitely many Ds and/or Cs (oo^0 = C XOR D = 1).
{___} = Cinf = Infinitely long C (0^0 = 1).

0^0 = oo^0 = 1 and we can see that we can't distinguish between C and D by their quantitative property.

But by their Structural property Dinf is not Cinf.

From the above we can learn that the Structure concept has more information than the Quantity concept in Math language.

Any element that is under a definition like "finite or infinitely many ..." cannot be anything but a member of D or Dinf sets, which have the structure of the Discreteness concept.

So, any line's segment is not a container but a connector between any two points {.___.}, and you can find this state in any scale that you choose.

C XOR D, and through this approach you don't have any contradiction between the Discreteness and the Continuum concepts, because any point is not in the Continuum, but an event that breaks the Continuum.

The Continuum does not exist in this event (because of the XOR between any line to any point), but any two events can be connected by a Continuum.

Take for example, the end of a line is an event that breaks the line and it turns to a Nothingness, so from one side we have the Continuum, from the other side we have the Nothingness, and between them we have a break point, that can be connected to another break point that may exist on the other side of the continuous line.
Another way to look at these concepts is:

Let a Continuum be an infinitely long X-axis.

Let a point be any Y(=0)-axis on the X-axis.

So what we get is a non-localized X-axis and infinitely many Y(=0)-axises points on (not in) the X-axis.

Through this point of view, the X-axis is a connector (not a container) between infinitely many Y(=0)-axises events.

In general, there are two levels of XOR:
A) ({} XOR {.}) OR ({} XOR {_})
B) {.} XOR {_}

There are 4 important conclusions from the above:

For example, let n = 3 = 1+1+1

A) 0^0 = Continuous 1

B) 1^0 = 1 Connector

C) n/1^0 = n Connectors (.__.__.__. = 1 1 1)

D) n/0^0 = Continuous n (.________. = 3)

Through this approach, each natural number is the associations (AND connective) between its continuous side (Continuous n) to its discrete side (n Connectors).

So from my point of view, Mathematics is more than variety of systems, where each system has its consistent universe.

As I see it, through this attitude one of the most important things of the evolution is cut out of today's Math.

Any evolution is based at least on two principles, variety and mutation.

The meaning of a mutation is to redefine existing things or familiar terms.

As I see it, the Modern Math has to look at this kind of approach as if it is a mutation in the Continuum concept and not as another axiomatic system where we can get:

|Q| < |R| = Pointed/segmented Continuum < A Pointless/segmentless Continuum .

Pointed/segmented Continuum is simply a contradiction, so as this concept does not exist, we can get a simple solution to the CH problem where:

|Q| < |R| < Continuum(={__}) .

 

[Organic]

The algorithm (which is based on Cartesian product) that was found and programmed by Stratman (by the way he call to these elements Sumbers,which is a combination of a set and a number) can easily find the associations products between {._.} forms and {.} forms:

For any n > 0, the following algorithm calculates the number of transitional states:

ASSOCIATION_LEVEL(int n):Vector

1 V:= new vector
2 if n = 1 then
3 V:=V + the single root of level 1
4 if n = 2 then
5 V:=V + the two roots of level 2
6 else
7 if n >2 then do
8 For each partition vector P in n do:
9 ROOTS(P,V)
10 return V

ROOTS(vector P,vector V)

1 roots :=[]
2 for each element in P do:
3 if element is a root then
4 roots:= roots + element
5 else
6 roots:=ASSOCIATION_LEVEL(integer value of element)
7 return horizontal alignment of the Cartesian product of P and draw new arc around them

You can check it here:

http://cyborg2000.xpert.com/ctheory/

Please don't go beyond 6 or 7.


The Cartesian product adding some left-right forms that can be ignored.

After Ablert Einstein there is no meaning to talk about space without time and vise versa, so any product is space/time protuct.

Through my point of view there is no meaning to talk about quantity without structure and vise versa, so any product is a structure/quantity protuct.

General conclusion:

The internal structure of any given quantity (finite or infinite) cannot be ignored.