Are Non-Commutative Natural Numbers the Future of Mathematical Theory?

[Hurkyl]

But what if I wanted to compute ((1*2)*(1+1))? Normally, I'd get 4. Are you saying this expression cannot be handled by your system?

 

[Organic]

By my system we have ((1*2)+(1+1)).

Again my system is more sensitive that the conventional system that cares only for the quantitative result and omit the structural difference between the elements.

Please show me the difference between ((1*2)*(1+1)) and ((1*2)+(1+1)) by the convetional system.

 

[Hurkyl]

Please show me the difference between ((1*2)*(1+1)) and ((1*2)+(1+1)) by the convetional system.

The first expression has two multiples and an add, and the second expression has two adds and a multiply. Their parse trees would look like

((1*2)*(1+1))
   *
  / \
 *   +
/ \ / \
1 2 1 1

((1*2)+(1+1))
   +
  / \
 *   +
/ \ / \
1 2 1 1

And what about ((1*3)*(1+1))? Does that also not exist in your system? Note that, even if we're only concerned with the end result, ((1*3)*(1+1)) and ((1*3)+(1+1)) are different.


I have another question; in your system, how come we can't add by 2, and get something like (1+2) being a distinct structure?

 

[Organic]

Hurkyl,

((1*2)*(1+1))
   *
  / \
 *   +
/ \ / \
1 2 1 1

((1*2)+(1+1))
   +
  / \
 *   +
/ \ / \
1 2 1 1

In both cases you used a private case of some structure in quantity 4.

My system define these structures as general ordered information forms that existing in any given quantity, and only then each information form can be used in many ways, which one of them is the way you used it.

Again, in the first stage my system defines the ordered information forms that existing in any given quantity.

And only then they are used, but this time not just as arbitrary separated information forms, but as a part of ordered information forms, for example:

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

In your example you used shape (4,6).

 

[Hurkyl]

which one of them is the way you used it.

You just told me that ((1*2)*(1+1)) does not exist in your system. How can a parse tree then, be using your system? In fact, you said that you cannot multiply things that have different "internal information structure", but I can make a new expression by multiplying any two expressions.


And why can't you add by 2 in your system?

Or multiply two things that are both different from 1?

And what about ((1*3)*(1+1))?


And how is (2, 1) different from (2, 2)? And if there really is a difference, why don't we see something that looks like (3, 1) but with a similar difference?

 

[Organic]

Hurkyl,

You still don't understand me, so I'll try again.

You used this information structure:

    .
   / \
  .   .
 / \ / \
.  ..   .

This is a general information form and you used it by put your notations on it.

We can use this structure for infinitely many other purposes.

My system define the ordered universe of these information forms, and then
we can use them, but this time we can find the deep relations between them
Because thet are ordered, by my method.

 

[Hurkyl]

So, it seems your theory is not about +, *, or natural numbers; it is about binary trees.


I'm still puzzled about why some things are in your ETtable and others aren't. e.g. Why are these different:

| |
+-+--
|


| |
+-+
|

and why don't you have one like:

| | |
+-+ |
|   |
+---+---
|
|

?


P.S. the '+' symbols are diagrammatic; I don't mean for them to be labels or placeholders or anything.

 

[Organic]

By the way, you used the wrong information form, instead you have to use:

          b  b      
          #  #              
   {a, b, a, a}   
    .  .  .  .       
    |  |  |  |     
    |__|  |__|_ 
    |+    |*        
    |     |         
    |     |         
    |_____|____
    |   *                  
    {{{x},x},{x,x}} 
 
          b  b      
          #  #              
   {a, b, a, a}   
    .  .  .  .       
    |  |  |  |     
    |__|  |__|_ 
    |+    |*        
    |     |         
    |     |         
    |_____|____
    |   +                  
    {{{x},x},{x,x}}

and why don't you have one like:

| | |
+-+ |
|   |
+---+---
|
|

See by yourself:

[b]1[/b]
(+1) = {1}

[b]2[/b]
(1*2)    = {1,1}
((+1)+1) = {{1},1}

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

[b]4[/b]
(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)
[b]5[/b]
...

 

[Organic]

So, it seems your theory is not about +, *, or natural numbers; it is about binary trees.

1) It is about the information forms that existing between the integral side and the differential side of any given natural number (in the first stage).

2) Binary trees are private cases in my universe.

 

[Hurkyl]

| | |
+-+ |
|   |
+---+---
|
|

is not an ETtree. Please see for yourself:

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

1) It is about the information forms that existing between the integral side and the differential side of any given natural number (in the first stage).

2) Binary trees are private cases in my universe.

But you said:

You used this information structure:

    .
   / \
  .   .
 / \ / \
.  ..   .

This is a general information form and you used it by put your notations on it.

We can use this structure for infinitely many other purposes.

My system define the ordered universe of these information forms, and then
we can use them, but this time we can find the deep relations between them
Because thet are ordered, by my method.

 

[cookiemonster]

http://www.physlink.com/Education/AskExperts/ae329.cfm

Schrodinger's Equation is based on a wave picture of QM.

Yes, thank you. I know what Schrodinger's Equation is.

My new nutural numbers are like w...ogen atom with your numbers? cookiemonster

 

[Organic]

Hi cookiemonster,

At this stage my system is a "pure" mathematical system.

If you understand the ordered redundancy_AND_uncertainty information forms model, then please use its products by yourself.

 

[cookiemonster]

So, at this instance in time but maybe or maybe not at some instance in the future, your numbers are 100% useless?

Good enough for me. Good day!

cookiemonster

 

[Organic]

So, at this instance in time but maybe or maybe not at some instance in the future, your numbers are 100% useless?

By your current understending of my system, the answer yes.

 

[cookiemonster]

"By my current understanding"? I have no understanding of your system. You have given me no reason to attempt to understand your system. I'm not against trying to understand your system if there is a reason, but you, the expert on and promoter of your system, have yet to provide one.

cookiemonster

 

[Organic]

cookiemonster,

My system is an ordered collection of infinitely many information forms that are ordered by their clarity degrees.

Shortly speaking, we have a Mendeleyev-like table of ordered information forms.

If we use these information forms as pert of our system, we get two benefits:

1) Local benefit: We have a concrete model of information form that we can research.

2) Global benefit: Because this information form belongs to an ordered universe, we can find the deep relations with another systems that are using these information forms as an "organic" part of them.

 

[cookiemonster]

My system is an ordered collection of infinitely many information forms that are ordered by their clarity degrees.

That's nice.

Shortly speaking, we have a Mendeleyev-like table of ordered information forms.

Again, that's nice.

If we use these information forms as pert of our system, we get two benefits:

Now we're getting somewhere.

1) Local benefit: We have a concrete model of information form that we can research.

A "model of information," huh. Really. "That we can research." Great, so I can figure out how many i's are in "information." Or maybe how many syllables are required to say "concrete."

A model of what kind of information? A model from which I can research what kind of knowledge? What kind of uses are we looking at? Your entire sentence is so cryptic and imprecise that it has no information.

2) Global benefit: Because this information form belongs to an ordered universe, we can find the deep relations with another systems that are using these information forms as an "organic" part of them.

"This information form." You still haven't mentioned what information form "this" is.

"An ordered universe." I don't believe our universe is perfectly ordered, and I'd wager that Quantum Theory agrees with me. Even when I scanned your webpage, I saw lots of mention of uncertainty.

"Deep relations." What?

"Another systems." Such as?

"'Organic' part of them." What?

What do these two reasons mean? Are they even saying anything? I mean, I'm not hard to fool here. I'm relatively uneducated in general and particularly unversed in your theory in specific. You could probably make something up that sounds good, is on the surface consistent, and makes sense and I'd buy it. But all you've said so far is that your system should be investigated "because it has things that should be investigated," which is hardly convincing.

Have you even arrived at any results, or have you been too busy drawing little diagrams? There's not much that's more convincing than results. If it works and I don't know why, then I'm relatively inclined to figure out why it does. But all your system is doing is sitting there doing nothing but stirring up matt and hijacking threads, and I don't know why, and I have no inclination to figure out why.

What's its purpose? What are you trying to accomplish with this? Are you striving toward some kind of goal? Have you gotten anywhere near that? Does your system satisfy that goal? If so, can we see this?

And for the third time, can you take any physical model of reality that has proven to be only an approximation, apply your numbers to it, and yield another physical model of reality that is a better approximation?

cookiemonster

 

[Hurkyl]

Perhaps you would care to draw a "redundancy / uncertainty" diagram for (5, 10)?


Why does "redundancy / uncertainty" never look like:

M  R  D
D  M  R

?

And what about ((1*3)*(1+1))?

 

[Organic]

Hurkyl,

My system is very accurate, and I show examples of it.

Plrease show me how can you define ((1*3)*(1+1)) when in my system

(1*3) means {1,1,1} and (1+1) means {{1},1}

Why does "redundancy / uncertainty" never look like:

M  R  D
D  M  R

Please give an example by using the lows of my system.



Here is again examples of my system, and this time try to understand my game:


Let # be XOR

My system is an ordered collection of redundancy_AND_uncertainy information forms.

See by yourself:

[b]1[/b]
(+1) = {1}

[b]2[/b]
(1*2)    = {1,1}
((+1)+1) = {{1},1}

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

[b]4[/b]
(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)
[b]5[/b]
...

also I showed that you used the wrong information form:

    b  b  b  b      
    #  #  #  #              
   {a, a, a, a}   
    .  .  .  .       
    |  |  |  |     
    |__|_ |__|_ 
    |     |         
    |     |         
    |     |         
    |_____|____
    |                      
    {{x,x},{x,x}}

instead you have to use:

          b  b      
          #  #              
   {a, b, a, a}   
    .  .  .  .       
    |  |  |  |     
    |__|  |__|_ 
    |+    |*        
    |     |         
    |     |         
    |_____|____
    |   *                  
    {{{x},x},{x,x}} 
 
          b  b      
          #  #              
   {a, b, a, a}   
    .  .  .  .       
    |  |  |  |     
    |__|  |__|_ 
    |+    |*        
    |     |         
    |     |         
    |_____|____
    |   +                  
    {{{x},x},{x,x}}

Each part of the graphic representation of it has an exact meaning, for example:

CR is Computational Root.

[u]A general graphic description of a CR[/u]

CD is Continuum AND Discreteness

RU is Rudandancy_AND_Uncertainty

AL is Association-Level

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

[url]http://www.geocities.com/complementarytheory/CATheory.pdf[/url] (page 7 - Indroduction (in the paper, not in the acrobat screen)).

You asked:

and why don't you have one like:

| | |
+-+ |
|   |
+---+---
|
|

My answer is:

If you use my system you have to follow my definitions, otherwise we are not talking about my system.

This form:

| | |
+-+ |
|   |
+---+---
|
|

has no meaning in my system.

For example let us say that there is a piano with 3 notes and we call it 3-system:

DO=D , RE=R , MI=M

The highest unclear information of 3-system is the most left information's-tree, where each key has no unique value of its own, and vice versa:

<-Redundancy->
    M   M   M  ^<----Uncertainty
    R   R   R  |    R   R
    D   D   D  |    D   D   M       D   R   M
    .   .   .  v    .   .   .       .   .   .
    |   |   |       |   |   |       |   |   |
3 = |   |   |       |___|_  |       |___|   |
    |   |   |       |       |       |       |
    |___|___|_      |_______|       |_______|
    |               |               |

 

[matt grime]

Hurkyl,
My system is very accurate, and I show examples of it.
Plrease show me how can you define ((1*3)*(1+1)) when in my system
(1*3) means {1,1,1} and (1+1) means {{1},1}
Please give an example by using the lows of my system.

lows? Perhaps you are self aware after all.

Here is again examples of my system, and this time try to understand my game:

we can't because you won't explain what your "game" is or does.

This form:

| | |
+-+ |
|   |
+---+---
|
|

has no meaning in my system.

What, and the other things you write do mean something?

 

[pallidin]

Organic, are you proposing an advanced system of information science?
If so, OK, we're listening.
May I suggest that you take the admittedly difficult time to show validation and usefulness. That is, could you show a concrete example where your system is superior than the one(s) currently used. A side-by-side definitive comparison would be helpful, and indeed essential.

 

[Organic]

Hi pallidin,

First, thank you very much for your positive attitude.

I have learned during the last year that my simple idea about the ordered universe of information forms, which can help us to define the deep connections between, so called, different systems, cannot easily be understood by professional "pure" mathematicians.

Maybe at this point I have to think about applied mathematics, but I need help for my first steps.

Please first, read this paper of mine:

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

Thank you,

Organic

 

[cookiemonster]

So are you saying that you have yet to find a concrete use for your numbers and need help finding one?

cookiemonster

 

[Hurkyl]

Plrease show me how can you define ((1*3)*(1+1)) when in my system

I have no clue how to define anything in your system.

It's easy enough to define it in ordinary mathematics, though.

Please give an example by using the lows of my system.

An example of what?

 

[Organic]

So are you saying that you have yet to find a concrete use for your numbers and need help finding one?

Yes, I need your help.

 

[matt grime]

I decided to check your score on the crackpot index. I'm doing this from memory of your posts. You can check this through jon baez's website. you score amazingly.

1 point for every statement that is widely agreed to be false (well, I reckon we'll go for 3 being generous) 3 points per statement that is logically inconsistent (you're scoring quite highly in the maths forum at the moment) then there's lots of intermediate ones but the peach has to be the end ones where you sail into the lead:

10 points for arguing that while a current well-established theory predicts phenomena correctly, it doesn't explain "why" they occur, or fails to provide a "mechanism".

10 points for each statement along the lines of "I'm not good at math, but my theory is conceptually right, so all I need is for someone to express it in terms of equations".

10 points for each new term you invent and use without properly defining it. (and let's face it this puts you off the scale)

10 points for claiming that your work is on the cutting edge of a "paradigm shift".

40 points for claiming that the "scientific establishment" is engaged in a "conspiracy" to prevent your work from gaining its well-deserved fame, or suchlike. (this one's arguable but I left out the one about saying how long you'd been working on the theory)

# 40 points for claiming that when your theory is finally appreciated, present-day science will be seen for the sham it truly is. (30 more points for fantasizing about show trials in which scientists who mocked your theories will be forced to recant.)

# 50 points for claiming you have a revolutionary theory but giving no concrete testable predictions.

those last two.. well I've not seen anyone take both of those awards before.

 

[Organic]

Matt,

And what you haveto say about that?:

If we use again the example of transformations between multisetes and "normal" sets, we can see that the internal structure of n+1 > 1 ordered forms, constructed by using all previous n > 1 forms:

[b]1[/b]
(+1) = {1}

[b]2[/b]
(1*2)    = {1,1}
((+1)+1) = {{1},1}

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

[b]4[/b]
(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)
[b]5[/b]
...

 

[Organic]

Dear Hurkyl,

I want to correct my previous answers to your questions.

Standard Math using only the 0_redunduncy_AND_0_uncertainty information forms.

For example, the information form, which is used in ((1*3)*(1+1)) is:

    a  b
    .  .        
    |  |      
    |__| 
    |

and when this building-block repeating within itself in two level, we get:

    1  3  1  1   
    .  .  .  .       
    |  |  |  |     
    |__|  |__| 
    |*    |+        =      ((1*3)*(1+1))   
    |     |         
    |     |         
    |_____|
    |  * 
    |

So, as you can see, my system is a first-order system of information forms, which existing within any given n.

Shortly speaking, first we define the information forms building-blocks, for example:

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

and only then we can use these building-blocks to construct our model.

Standard Math using only the last form of each collection that existing within any given n, for example:

      [b]0[/b]
      [b].[/b]
1 =   [b]|    
      *[/b]


    1   1
    0   0     [b]0   1[/b]
    .   .     [b].   .[/b]
    |   |     [b]|   |[/b]
2 = |___|_    [b]|___|[/b]
    |         [b]| *[/b]


    2   2   2
    1   1   1       1   1
    0   0   0       0   0   2       [b]0   1   2[/b]
    .   .   .       .   .   .       [b].   .   .[/b]
    |   |   |       |   |   |       [b]|   |   |[/b]
3 = |   |   |       |___|_  |       [b]|___|   |[/b]
    |   |   |       |       |       [b]|       |[/b]
    |___|___|_      |_______|       [b]|_______|[/b]
    |               |               [b]|   *[/b]

    
    -------------->>>
    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           1  1
    0  0  0  0     0  0  0  0     0  1  0  0     0  0  0  0     0  1  0  0
    .  .  .  .     .  .  .  .     .  .  .  .     .  .  .  .     .  .  .  .
    |  |  |  |     |  |  |  |     |  |  |  |     |  |  |  |     |  |  |  |
    |  |  |  |     |__|_ |  |     |__|  |  |     |__|_ |__|_    |__|  |__|_
    |  |  |  |     |     |  |     |     |  |     |     |        |     |
    |  |  |  |     |     |  |     |     |  |     |     |        |     |
    |  |  |  |     |     |  |     |     |  |     |     |        |     |
    |__|__|__|_    |_____|__|_    |_____|__|_    |_____|____    |_____|____
    |              |              |              |              |
 
4 =                2  2  2
                   1  1  1        1  1
    0  1  0  1     0  0  0  3     0  0  2  3     [b]0  1  2  3[/b]
    .  .  .  .     .  .  .  .     .  .  .  .     [b].  .  .  .[/b]
    |  |  |  |     |  |  |  |     |  |  |  |     [b]|  |  |  |[/b]
    |__|  |__|     |  |  |  |     |__|_ |  |     [b]|__|  |  |[/b]
    |     |        |  |  |  |     |     |  |     [b]|     |  |[/b]
    |     |        |__|__|_ |     |_____|  |     [b]|_____|  |[/b]
    |     |        |        |     |        |     [b]|        |[/b]
    |_____|____    |________|     |________|     [b]|________|[/b]
    |              |              |              [b]|   *[/b]

The bold forms that notated by * are number system representations, based on Peano axioms (Standard Math information forms).

All the other first-order information forms, are new forms, which are not used (yet) by Standard Math language as building-blocks of natural numbers.

Why does "redundancy / uncertainty" never look like:

    M   R   D    
    D   M   R    
    .   .   .      
    |   |   |     
    |   |   |     
    |   |   |     
    |___|___|_   
    |

A complete state of Redundancy_AND_uncertainty within quantity 3 cannot be less then 3 different possibilities for each discrete element.

If we use again the example of transformations between multisetes and "normal" sets, we can see that the internal structure of n+1 > 1 ordered forms, constructed by using all previous n > 1 forms:

[b]1[/b]
(+1) = {1}

[b]2[/b]
(1*2)    = {1,1}
((+1)+1) = {{1},1}

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

[b]4[/b]
(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)
[b]5[/b]
...