A gate to complexity

1. Mar 27, 2004

Organic

Let us say that we wish to construct and explore a non-trivial numerical form of structural/dynamic abstract or non-abstract complex models.

If there was a way to choose the unique building blocks that we need for this goal, it was our gate to the universe of complexity exploration.

For this goal we need at least to properties:

1) A quick way to find any needed complex building block.

2) An ordered collection of unique complex building blocks.

Let us say that we want to construct a model of an information tree, where each building-block is a sub-tree that represents information clarity degree.

The result of this construction is a very complex model of finite or infinitely many ordered or unordered levels of information's clarity degrees upon finite or infinitely many different scales.

An example of (1) and (2):

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

An example a complex model can be found here:

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

Please be aware that the Natural numbers are used here as a private case of the clearest information tree, which is used here to find the exact x,y location of each unique building block, but it does not give us any information on the internal unique structure of this building block.

More than that, we cannot use the natural numbers to quantify the internal structure of any given buiding-block, which is based on uncertainy_AND_redundancy > 0, because the value of n>0 in this case, is a general representation of several random and different possibilities between 0 and k>=n.

Shortly speaking, our building-blocks locations can be found by N members, but their internal structure cannot described by them, because N members internal information structure, is based on 0 redundancy AND 0 uncertainty information form.

An example:

Let us take the trees that included in quantity 3:

{1,1,1} , {{1,1},1} , {{{1},1},1}

The first is a multiset and the last is a "normal" set.

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

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 = |   |   |       |___|_  |       |___|   |
|   |   |       |       |       |       |
|___|___|_      |_______|       |_______|
|               |               |

An example of 4-notes piano (DO=D , RE=R , MI=M , FA=F):

------------>>>

F  F  F  F           F  F           F  F
M  M  M  M           M  M           M  M
R  R  R  R     R  R  R  R           R  R     R  R  R  R
D  D  D  D     D  D  D  D     D  R  D  D     D  D  D  D
.  .  .  .     .  .  .  .     .  .  .  .     .  .  .  .
|  |  |  |     |  |  |  |     |  |  |  |     |  |  |  |
|  |  |  |     |__|_ |  |     |__|  |  |     |__|_ |__|_
|  |  |  |     |     |  |     |     |  |     |     |
|  |  |  |     |     |  |     |     |  |     |     |
|  |  |  |     |     |  |     |     |  |     |     |
|__|__|__|_    |_____|__|_    |_____|__|_    |_____|____
|              |              |              |

4 =
M  M  M
R  R                     R  R  R        R  R
D  R  D  D      D  R  D  R     D  D  D  F     D  D  M  F
.  .  .  .      .  .  .  .     .  .  .  .     .  .  .  .
|  |  |  |      |  |  |  |     |  |  |  |     |  |  |  |
|__|  |__|_     |__|  |__|     |  |  |  |     |__|_ |  |
|     |         |     |        |  |  |  |     |     |  |
|     |         |     |        |__|__|_ |     |_____|  |
|     |         |     |        |        |     |        |
|_____|____     |_____|____    |________|     |________|
|               |              |              |

D  R  M  F
.  .  .  .
|  |  |  |
|__|  |  |
|     |  |
|_____|  |
|        |
|________|
|

I like to look at the connection between redundancy_AND_uncertainty as "a cloud of possibilities", you know like the possibilities that we have in some Quantum element in the same space/time.

For example:

Let C be a closed door.

Let O be an opened door.

Let # be XOR.

Code (Text):

C   C
#   #
O   O
.   .
|   |
|___|_
|

O   C
.   .
|   |
|___|
|

When we are using N members to describe information form which its redundany_AND_uncertainty levels are greater than 0, the best we can do is to use multiplication and addition as complementary operations, for example please look at page 7 (in the paper, not in the acrobat screen):

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

Now, look at this:

through this structural/quantitative point of view 1*5 not= 1+1+1+1+1 not= 1*3+2 not= 1+4 and so on, because each arithmetical expression has a unique information form.

I did not find any mathematical branch that distinguishes between arithmetic operations according to their level of clarity, for example:

1*5 = *5 = {1,1,1,1,1}
1*3+2 = *3+2 = {{{1,1,1},1},1}
1*3;+2= *3;+2 = {{1,1,1},{{1},1}}
1*3;1*2= *3;*2 = {{1,1,1},{1,1}}
1+1+1+1+1 = +5 = {{{{1},1},1},1},1}

and so on ...

Shortly specking, any use of N to describe an information building-block, is at least some n AND addition or multiplication operations.

A proof that cannot be done by using standard N members.

Theorem: 1*5 not= 1+1+1+1+1

Proof: 1*5 = {1,1,1,1,1} not= {{{{1},1},1},1},1} = 1+1+1+1+1

Shortly specking, any use of N to describe an information building-block, is at least some n AND addition or multiplication operations.

Last edited: Mar 28, 2004
2. Mar 28, 2004

Organic

By my point of view -, +, and * have two worlds, the internal world and the external world.

In standard Math the Natural numbers world is only the external world, where each operation chenging the quantity of n.

Let us look on - and + operations on n from ET eyes:

The external result of ((((1),1),1),1) - 1 is (((1),1),1)

The internal result of ((((1),1),1),1) - 1 is (((1,1),1),1)

The external result of (((1),1),1) + 1 is ((((1),1),1),1)

The internal result of (((1,1),1),1) + 1 is ((((1),1),1),1)

So as you see - and + do not changing the quantity but the symmetry degree of each ET.

In this case (1,1,1,1) is the maximum symmety degree and ((((1),1),1),1) is the minimum symmetry degree, for example:

Let us say that we have here a transformation between
multiset {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:

Code (Text):

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 see the piano model in:
http://www.geocities.com/complementarytheory/HelpIsNeeded.pdf

General conclusion:

The internal structure of any given quantity (finite or infinite) cannot be ignored, therefore the natural numbers are at least structural/quantitative information forms:
http://www.geocities.com/complementarytheory/ETtable.pdf

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

Let us say that in the first stage we get an ordered table of infinitely many symmetry forms, and by this ordered table we can start to explore the relations between highly complex different forms of symmetries.

Shortly speaking, we have in our hand a Mendeleiev-like table of symmetries, where Peano axiom symmetries, are only some private case in it.

Another very interesting thing is that from ET point of view any number is an organic form with internal complexity, that can exist iff we take each ET as a whole, which is a paradim shift in the concept of a NUMBER.

And this paradigm shift is based on this simple test:

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

Any number is first of all an information form, therefore any aspect of information form MUST be researched by us, where our cognitionâ€™s abilities to research information MUST be included too.

Form this point of view, redundancy AND uncertainty cannot be ignored, and through this approach(which is not an extra approach but the MINIMAL approach to understand the natural number concept) we can clearly show that the standard natural numbers are only a one and only one private case of verity of information forms, which are ordered by their vagueness degrees from maximum vagueness to minimum vagueness when a given quantity remains unchanged.

Man is no longer an observer but a participator, which its influence must be included in any explored system.

The above is the QM paradigm shift that is not understood yet by the current community of pure mathematicians.

For example: Be aware that what you call a function is first of all a reflection of your memory.

3. Mar 28, 2004

moshek

Organic:

Thank you for the new represantation to your non-conventional attitude to mathematics in general.

I want to ask you the folowing question:

What is not working in regular mathematics in the way we see complexity
so we need acording to your new thread to learned how to count in new way that you suggest us?

Thank you
Moshe

4. Mar 28, 2004

Organic

x = a XOR b XOR c XOR ...

It means that x value is one and only one result out of several possibilities.

x is a one result of many possibilities.

By this way 1 has two opposite forms.

Form-A is the standard meaning of 1 as a one well-known value.

Form-B is my new 1, where 1 stands for a one and only one result out of several possibilities.

An example of form-A set: {{{1},1},1}.

An example of form-B set: {1,1,1}

5. Mar 28, 2004

moshek

So set can contain the same element few times ?

6. Mar 28, 2004

Organic

Form-A set is a "normal" set, where Form-B is what standard Math calls a "multiset" ( http://mathworld.wolfram.com/Multiset.html ).

My natural(organic) number is based on fading transition between "normal" set and "multiset" and vise versa.

In this fading transition between the opposites Form-A and Form-B sets,
mutiplication can be oprated only among Form-B members, where addition can be operated among Form-A members and also among Form-B members.

Last edited: Mar 28, 2004
7. Mar 28, 2004

moshek

Organic :

1) A positive solution to GCH !
2) Set theory without axiom of choice !

What is really the value of your new paradigm on Set theory as you describe here.since you suggest us to start everting from the begining when we count numbers.

Thank you
Moshek
:wink:

Last edited by a moderator: Apr 20, 2017
8. Mar 28, 2004

Organic

Sets or numbers are first of all information forms.

Redundancy and uncertainty are fundamental concepts of any information form.

It means that they must be included right from the "First order" level of Math language.

The result of this attitude is a new interpretation of Set and Natural number concepts, as clearly can be shown in my structural/quantitative approach to these two basic concepts.

Last edited: Mar 28, 2004
9. Mar 28, 2004

moshek

You built the uncertainty principle of Quantum mechanics right
in the foundation of all mathematics the concept of number.

So how is that relate to Prague work on the establish of first order logic.

10. Mar 28, 2004

Organic

11. Mar 28, 2004

moshek

Organic :

As I can see your work expand Frege work since in his presentation there in no redundancy when he develop first order logic by aritmetic. While you add it naturally, this is very nice !