Organic Numbers

[matt grime]

In semi-response to Organic's post I thought I'd half take up one of his challenges:

Let S = NxT be the product of the natural numbers, N, with the set of all rooted finite trees (or directed graphs satisfying the obvious conditions), embedded in the plane, with the natural ordering on the branches/leaves.

Let C be the subset of all possible { (n,t) | n in N, t a tree with exactly n edges}

We define an operation + on elements of c:

(n,t)+(m,r) = (n+m,s) where s is the tree obtained by gluing the tree r onto the end of the first leaf.

The subset (n,t) with t the trivial tree with 1 leaf and n edges, forms a copy of N under addition.

We define * to be (n,t)*(m,s) by

(((...(((n,t)+(n,t))+(n,t))+...)+(n,t))

where there are m-1 addition signs.

again (n,t) with t the tree with n edges and 1 leaf, is a subset that forms a copy of N under multiplication.

neither + nor * are in general commutative, and I doubt they are associative either, but I can't be bothered to check, they are both well defined binary operations from CxC to C.

Now shall I claim that C is the new non-commutative natural numbers or not?

[Organic]

but he's already using the naturals inside his definition to define his "new naturals" there is nothing there to suggest these new objects should replace the natural numbers.

So, you dont understand that the standard natural number is a trivial private case of infinitely many structural/quantitative information's forms that ignored by Standard Math paradigm.

Do you get it?

A NUMBER is first of all an information's form, and to understand this we MUST explore our cognition's abilities to define this information's form, as I do here:

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

[moshek]

Dear Matt:

I am really happy that you did so !
I will come back only in this weekend.

Best
Moshek
[:))]

[matt grime]

Originally posted by Organic
So, you dont understand that the standard natural number is a trivial private case of infinitely many structural/quantitative information's forms that ignored by Standard Math paradigm.

Do you get it?

A NUMBER is first of all an information's form, and to understand this we MUST explore our cognition's abilities to define this information's form, as I do here:

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

Nothing to do with my post here then?

When you, in your theory, assign these trees to certain sets, how do you decide which belong in the same set? You do it by counting the things involved in the construction, therefore you are explicitly using the counting numbers. Otherwise how did you decide how to group certain things together?

Nothing to say about my construction here then? This is the answer to your request ot show you something that behaves like your structures do. The difference is I don't make any large and unjustified claims about what I'm doing.

[moshek]

Well I am still here.

Organic:

Matt open a great opportunity here
Please see this as a real challenge for you.

Yours
Moshek[:))]

I will come back in few days.

[matt grime]

I'm not in the slightest against you putting extra structure on the natural numbers, I just would like you to realize that you don't need to, and in fact you are requiring their 'non-existent' quantity only property when you describe the extra structure. The natural numbers come only as the idea of quantity, even their additive and mulitplicative structure arises from this.

It is not your ideas that annoy me; it is your mistaken assertions about them and mathematics.

The philosophical aspects of it are of minimal importance, especially until such time as you even begin to demonstrate you understand what you write.



Here let me demonstrate a rigorous approach using the things I defined above.

Define the complexity of (n,t) to be the number of leaves of t - this is equal to the number of paths from the root to the end of a branch, by the definition of tree.

If all paths from root to branch have the same length, let us define the symmetry thus:

Each set of consecutive leaves, take the smallest subtree containing them, call this a cutting. If the tree can be separated into r isomorphic cuttings that only intersect at the root, we say it has symmetry degree r.

A tree with different length paths we call asymmetric.

Characterization of N inside C: it is the set of elements of symmetry degree 1 with path lengths maximal among (n,t) for each fixed n.

Moreover it is equal to the subset of elements of complexity 1.

That's how you write maths that other people can understand. You use terms everyone knows, or can look up knowing it will agree with your terminology. Anything new you define so that it is unambiguous.

If we denote the degree of (n,t) as n.
Then n is prime iff all trees of degree n are of complexity 1 or asymmetric.


How's that for you? Is this the kind of thing you had in mind when you cahlleneged us to find some ordinary mathematics that does what yours does? As you've never even shown us what yours does of course it's a little tricky.

[Organic]

Matt Grime,

Nothing to say about my construction here then? This is the answer to your request ot show you something that behaves like your structures do. The difference is I don't make any large and unjustified claims about what I'm doing.

Your construction has nothing to do with my natural numbers because,
by your definitions, uncertainty AND redundancy are ignored and also the complementary relation between multiplication and addition.

To understand it better please look at my ET's and find by yourself
that the Equations Trees can be changed by their structure, when quantity remains unchanged:
http://www.geocities.com/complementarytheory/ET.pdf

Please show me this property in the standard natural numbers, where each change is a quantity change (because any change in no-redundancy_no-uncertainty information form immediately changes its quantity).

I don't make any large and unjustified claims about what I'm doing.

You right because you don't know what you are doing.

Let S = NxT be the product of the natural numbers, N, with the set of all rooted finite trees (or directed graphs satisfying the obvious conditions), embedded in the plane, with the natural ordering on the branches/leaves.

Why you so afraid to understand that the "quantity-only" natural numbers are only partial information of the structural/quantitative information that I show in my system?

Do you really cannot understand the paradigm shift the QM gave to the scintific world?

Each one of these structural-quantitative products is unique, therefore can be used as a building-block for much more interesting and richer information form, then your “quantitative-only" unique [n] result, which is nothing but a private-case of no-redundancy-no-uncertainty structural-quantitative product of my number system.

We can clearly see this here:

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


The big paradigm's shift is QM and not SR, please read this:

http://plato.stanford.edu/entries/qm-copenhagen/#4

This paradigm's shift, does not exist in the basis of Standard Math language, because Boolean Logic or Fuzzy Logic are private cases of what I call Complementary Logic, that an overview of it can be found here: http://www.geocities.com/complementarytheory/BFC.pdf

Through my point of view Natural numbers are complementary elements, based on discreteness(particle-like)-continuum(wave-like) associations.

The information structure of the standard Natural numbers, is only a private case of these associations, for example:

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

More details can be found here:

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

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

It means that we cannot ignore our cognition's abilities to create Math language anymore, as I clearly show here:

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

[matt grime]

Definition:

1 minus the reciprocal of the complexity is the uncertainty of (n,t)

Therefore the natural numbers are a private case where there is no uncertainty

Demonstrate I am wrong.

I declare that my numbers are complementary objects that indicate the paradigm shift to a quauntum viewpoint of mathematics.

Demonstrate I am wrong.

Define the redundancy of (n,t) to be n minus the length of the longest path through the tree from root to branch.

The usual natural numbers are a 'private case' where the redundancy is zero.

I can remove all reference to N from the constructions involved in this if you feel like it too.

[Organic]

Define the redundancy of (n,t) to be n minus the length of the longest path through the tree from root to branch.

1)Redundancy AND uncertainty

2)Show now where is the tree?

(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)

3) You call t an Organic numbers but you don't understand what Are Organic numbers.

Organic numbers cannot reduced to quantity alone, because their organic information structure cannot be ignored.

It means that each information form is at least an organic stuctural/quantitative unique element as we can find here:
http://www.geocities.com/complementarytheory/ET.pdf

[matt grime]

I defined uncertainty and redundancy for my trees, which are not just quantity. Given any quantity, which I called degree there are many elements of that degree. We can label [n] as all the trees of degree n.

Show that my construction has less right to be called the correct interpretation of the new QM paradigm of mathematics? I've got uncertainty, redundancy, quantity, and operations that I call complmentary addition and multiplication. So why am I wrong and you right? They're all based on discrete and continuous constructions such as nodes as branches which encompass all the requirements of the continuum and such.

So I now contend that my number system is the correct one, yours but a pale attempt to obtain this level of complexity and accuracy.

Demonstrate I am wrong.

[Organic]

Matt,

It is very simple to show that you don't understand my system, for example:

Please find the unique labels of {1,1,1,1}.

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

[matt grime]

Originally posted by Organic
Matt,

It is very simple to show that you don't understand my system, for example:

Please find the unique labels of {1,1,1,1}.

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

What did I call function?

I'll will call the tree (4,t) where t is the tree with just the root node and 4 leaves from it) the element (1,1,1,1), or perhaps I want the trivial tree of degree 4? I can't remember what you're claiming (1,1,1,1) represents, tell me and I'll tell you which one it is.

I am not claiming my system is your system, but it that it has all the features of your system, with the added bonus of containing a genuine copy of N inside it, as well as a non-commutative deformation of it. And therefore has just as much right to be called the correct system of the 'natural numbers' of mathematics. It contains all possible tree structures of degree n, so it remembers the 'structure' of each number as well as its quantity. I can change the structure and not change its quantity (degree).

Looks like it's a winner.

Oh, and to make it more rigorous, I only defined symmetry degree for symmetric trees, so I will define the symmetry degree of an asymetric tree to be -oo,

[Organic]

BY writing (4,t) you did nothing.

Show us how this general (n,t) can define the number of (6,t) unique elements, for example.

[matt grime]

the number of elements in [n] ie the set of all (n,t) t a tree with n edges, is exactly the number of (rooted) trees with n edges. You can count them if you like.

There is one element of[1], 2 elements in [2], there are I believe 5 elements in [3]...., call the number of elements in [n] the rank of n:=r(n)

there are then (r(n),t) elements of degree n, where t is the trivial tree with r(n) edges and 1 leaf. Recall the trivial trees form the trivial N structure of quantity only.

Why, how many structures are there for each quantity in your theory?

Perhaps, if my answer didn't satisfy you, you could tell me what the question actually means because I don't understand what you want. None of the things you've asked me about have been defined in this theory using those words. Please exlpain what you want me to define now, and I'll endeavour to add it in.

For every question you ask, as well, could you show what the answer is in your system too, so we know that you're not just asking deliberately meaningless questions, which is how they appear as you don't adequately explain what you want.

[Organic]

Matt,

Here are numbers (1,t) to (5,t):
http://www.geocities.com/complementarytheory/ETtable.pdf

Find the rule and give the number of (6,t)unique elements.

[matt grime]

Originally posted by Organic
Matt,

Here are numbers (1,t) to (5,t):
http://www.geocities.com/complementarytheory/ETtable.pdf

Find the rule and give the number of (6,t)unique elements.

That question doesn't make sense in english. I can tell you there are as many elements in my set [6] as there are rooted trees with 6 edges, off the top of my head I don't know how many of those there are. There is probably a simple generating function for them but I can't be bothered to work it out. Is that waht you wanted to know? The rank of 6? Using my definition of rank as above?

Anyway, that diagram refers to your trees, not mine. I am not saying our systems are the same, I'm just saying that mathematically I can define a system properly with all the innate structure of yours but in a way that anyone can understand.

Try rewriting the question so it conforms to the basic rules of English and perhaps I can answer it.

And why are you using my notation for your objects?

[Organic]

And why are you using my notation for your objects?

You know what? show how you can find the number of trees of (6,t)
in your system.

[matt grime]

It's the number of trees with 6 edges. Get a pen and paper and work it out. It is tedious but doable, hint find the correct recusrive formula. Why, what are the number of diagrams in your theory with quantity 6?

[Organic]

67 unique ordered trees (by their symmetry degrees) for quantity 6,
without left-right switches.

But you see, what is important here is not just the quantity 67 but the unique structure of each tree.

By my research here:
http://www.geocities.com/complementarytheory/count.pdf

Any number is at least an association between our memory (the continuum) and some element(s)(discreteness).

Therefore these trees are the basic models of the associations between our memory and a given quantity.

The standard natural numbers are private case of one and only one association tree, which is the maximum broken symmetry tree of any given quantity.

Because any number is first of all memory AND element(s), then no association's structural form can be omitted by us, when we count.

Therefore N members of standard Math are trivial elements, and any other number system that constructed by them is also a trivial number system.

My theory of numbers fixes this triviality by exposing the hidden internal information structures of any given quantity.

Because my organic numbers are an ordered number system by symmetry degrees, we can use them as non-trivial powerful tools that can help us to start a non-trivial research of the complexity itself.

A simple example:

To say 2+3 is not enough because in my natural number system we need to know what internal structure of 2 and what internal structure of 3 we are going to add to each other.

[matt grime]

Quite possibly true, you are now into the philosophy of mathematics. There you are free to state what you will. I will not and have not argued against that position. I have always argued against you misusing mathematical terms, and not telling anyone what you mean clearly.

Here I've demonstrated a mathematical object that does all the things you want and that you claimed mathematics wasn't able to do. It possesses attributes labelled by all the terms you gave, has well defined binary operations (for all inputs, unlike yours) and is a genuine non-commutative extension of N, which it contains as a subobject.

Incidentally is 67 the answer in your system or mine?

The formula for the number of trees (rooted, ordered, etc) with n edges is I believe

sum r(i)r(j) where i+j=n-1 and i,j are non-negative, and r(0) is defined to be 1.


Which gives, r(1)=1, r(2)=2, r(3) =5, r(4)=5+2+2+5=14 r(5)=14+5+4+5+14=32 r(6)=32+14+10+10+14+32=112 I think.


In my system the Natural numbers are the subobject corresponding to the case of zero redundancy and zero uncertainty, just as you wished.

[Organic]

Matt,

I do not say that Math cannot do it, I say that Math not use these internal symmety degrees as the fundamental property of The natural number definition.

If we use these symmetry degrees as the basis of Natural number definition, then any Natural number become a much for powerful/sensitive tool for Math language research.

[matt grime]

But if you add this structure to the definition of natural numbers, then you are changing the definition, tautologically. This stucture is not needed to define the natural numbers, it can be deduced from their existence, and is therefore not a "first order" property of them. I use that in a non-technical sense. All the objects, trees etc, you use are out there already known and studied. You've given no compelling reason to justify the assertions you've made about why the natural numbers must have links to uncertainty, or quantum objects. Philosophically I believe you have the wrong emphasis, but that difference stems from your need to define mathematics subjectively and use this idea of being aware of its being observed. I don't see any mathematical reason for that. Philosophically that is a different matter; I have no strong opinions either way. But that is my personal philosophy of mathematical research, which is of no interest to anyone but me at the moment.

I just wish you would at least define unambiguously the terms you use so that misunderstandings cannot arise.

[Organic]

Matt,

Please show me some Mathematical research where trees as symmetry degrees are studied, including detailed structural results.

[Organic]

This stucture is not needed to define the natural numbers, it can be deduced from their existence, and is therefore not a "first order" property of them.

How do you know that these symmetry degrees are not "first order" property of them?

Show me some research that proves it.


For example: 6 gives, let us say, 67 unique ordered symmetry structures.

In 67 we are talking on very big quantitative result of unique ordered symmetry structures that are ignored by Standard Math point of view, and so on...

[matt grime]

If you explain what you mean by symmetry degree perhaps I could.

And I said that I wasn't using "first order" in its proper meaning. Intuitively, you are using the quantity idea to enumerate your 67 trees of quantity 6. If the (structureless) natural numbers weren't available how could you use your "natural numbers" to count your "natural numbers"...? Especially as the 'number' of trees of a given quantity is more than any of the quantities defined so far - you need to "define 67" to count the structures of "quantity 6" ... This isn't rigorous, nor do I intend to make it so; I'm offering a philosophical discussion of the ideas because that way none of us can be wrong, just disagree, possibly.

[Organic]

Matt,

please look at these Equation Trees(=ET) :



1 1 1
+ 2 = + 2 = +
1 1 1
4 = + 4 = + 4 = +
1 1 1
+ + 2 = +
1 1 1


1 1
+ 2 = +
3 = 1 3 = + 1
4 = + + 4 = +
1 1

1 1

Let us say that notation '4' is the integral side of ET4,
and '1 1 1 1' notations are the differential side of ET4.

Now, take each ET as a one organic element.

From this point of view the quantity of each ET remains
unchanged , but the structure of each ET is unique and it is not
ordered by quantity change but by the symmetrical degree of each ET.

Again we count the ET's not because of quantitative change, but because of
The inner structural change of each ET.

Now, we can find exactly 9 unique ET's in quantity 4:

These ET's can also be represented as:

(1,1,1,1) <------------ Maximum symmetry-degree,
((1,1),1,1) Minimum information’s clarity-degree
(((1),1),1,1)
((1,1),(1,1))
(((1),1),(1,1))
(((1),1),((1),1))
((1,1,1),1)
(((1,1),1),1)
((((1),1),1),1) <------ Minimum symmetry-degree,
Maximum information’s clarity-degree

By Peano axioms we are using only this ET (((((1),1),1),1),...) Which is a private case of one and only one ET structure.

But what about the rest ET's of ET4?
Any changing in ((((1),1),1),1) form cannot be but a quantitative change, because he has no "inner space" that can be changed.

But this is not the case of the rest ET's

(1,1,1,1) <------------ Maximum symmetry-degree,
((1,1),1,1) Minimum information’s clarity-degree
(((1),1),1,1)
((1,1),(1,1))
(((1),1),(1,1))
(((1),1),((1),1))
((1,1,1),1)
(((1,1),1),1)

that have "a room" for internal changes.

Any ET with "free space" for structural changes is ignored by standard Math.

By my structural/quantitative approach this information is not ignored.

1) we indeed using peano axioms to find the next basic quantity, then for each basic quantity we define all its inner structures, and then we move to the next basic quantity by using again Peanos axiom.

2) Through this attitude we systematically define any possible information structure That can be expressed by ET's.

3) Then we can research their relations upon infinitely many scales, and create more and more interesting models of complex information structures in very efficient ways.

As you can see here: http://www.geocities.com/complementarytheory/ETtable.pdf

The blue x axis is Peano world, and the magenta y axis is the rest of ET's world.

Please look again in these examples:

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

Also, please look again on this paper:

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


Thank you,

Orgainc

[matt grime]

I suppose a philosophical position moshek might approve of is that your constructions require more ontological commtiment than is necessary to define the natural numbers.

You need to have trees, symmetry (which you've still not explained) and indeterminacy (ditto) and the notions of very complex sets lying around. In fact your (well, stratman's) construction uses the langauge of computer programming (let n=1....) and so uses explicilty the natural numbers (as understood ordinarily). Of course that last one isn't a fatal problem in itself. So you not only require the existence of the natural numbers as counting objects, but further ideas, that require ontological commitment that is not necessary in classical mathematics.

[Hurkyl]

Here's a question.

Can you prove something about quantity that cannot be proven through usual mathematical methods?


Allow me to emphasize that I'm talking about purely quantitative statements, not statements where we've substituted your new concepts for existing concepts (e.g. your use of aleph0 seems vastly different from the "quantitative" use of aleph0)

[Organic]

Hyrkyl,


Can you prove something about quantity that cannot be proven through usual mathematical methods?

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:


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}


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

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}


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

Please read also this paper:
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 cognition’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.

[matt grime]

That was one of the longest 'no's I've seen.

Hurkyl asked you to use your theory to prove something abou quantity that cannto be proven with the usual concept of the natural numbers.

[Organic]

My answer was, first you have to understand that we are talking about new dimension of the natural number that its results cannot be reduced to quantity only picture.

My point of view is a comprehensive point of view on the concept of a number that can open a new unexplored yet mathematical dimension.

For example because you don't do this paradigm shift in your cognition, you cannot understand these proofs:

http://www.geocities.com/complementarytheory/3n1proof.pdf

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

Time after time I do my best to show you the new dimension that is based on:

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

Since you ignore the meaning of this simple test, you will see only the shadow of what I have to say, no more no less.

[matt grime]

The idea that, given a natural number, n, one can assign many structures to it that are more complex and richer is not a new idea. And until such time as you explain what the terms in your construction mean you won't win any plaudits.

[Organic]

Please show me a mathematical research about natural numbers that are ordered by their vagueness degrees from maximum vagueness to minimum vagueness when a given quantity remains unchanged.

Let # be XOR condition.

vagueness defined as: {a#b#c#..., a#b#c#... ,...}

[matt grime]

Only if you tell us what vague ness degree is.

[Organic]

vague ness is some n>1.

[matt grime]

Vagueness is some n>1?

Please elaborate because that is an ambiguous sentence, and I cannot make any of its potential meanings mathematical.

[Organic]

But again you miss the point because if you understand the meaning of the result of this test:

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

you have no choice but to agree with that that the minimal existence of the natural number cannot be less then structural/quantitative information form.

Symmetry and probability have to be used right from the most fundamental level of any mathematical system, and I mean from the level of the set concept, or form the natural numbers (what you call "first order").

The most simple element that can combine symmetry and probability, has an information form of a tree.

And when the natural number is a least a tree, all other number systems are also trees.

Let us do some pause for my little song:

Every number is a tree,
climb on it, its all for free,
full of branches, brids and winds,
no more losers no more wins.

Peace on earth
and in the sky
every heart
can bloom and fly.

[moshek]

A nice poem Organic !

In a very intersting Thread Matt !

With a fress new forum Formant Hurkel !

I am here again Moshek.
[t)]

[matt grime]

But I don't agree with your "test" (there's nothing there that is a test as far as I understand the word) as it is not mathematical, and is rather silly. What does it matter that we may or may not be allowed to "use our memory" whatever that may mean, nor do I see why you've negated whatever problem it is that you think is there. And that doesn't alter the fact that you have not told me what you mean be the sentence

'vagueness is any n>1'

[Organic]

...it is not mathematical,...

1) There is no such a thing like mathematics where our cognition's abilities
are not deeply involved within it.

2) Therefore any mathematical research MUST include our abilities to define and develop it, and this is exactly the meaning of my test.

3) In this test I show these important things:

a) Without an association between some element AND our memory, we cannot define any quantity beyond 1.

b) Because the minimal conditions to count beyond 1 are at least memory AND element associations, then any minimal information form beyond 1 MUST be the information form of a tree, where the root is our memory and the elements are its branches and/or it leafs.

4) Because when we count, we most of the time using our memory in a sequential way we (without paying attention) jumping straight to the information form of an ordered tree.

5) By checking the ordering process I clearly show that in any given quantity there are several stages (or satiations if you will) that can be defined and be ordered by their clarity degrees, where the first information tree represents the most unclear information and the last tree represents the most clear information.

6) These stages belong to the "first order" level because no information form of memory_AND_elements tree can be ignored.

[matt grime]

The numbers exist, you just don't remember they exist, might be a better way of saying it. If you cannot use your memory, how do you even know you were asked to count them in the first place? The question as to whether this has anything in slgihtest to do with mathematics is still open. How did you remember all the squiggly symbols you've just used to write that post? What if...?

[moshek]

I was passing yesterday near the bulding of the conference "Cardinals at work" :

1) Resolving the GCH positively ( Woodin, Shelah)
2) Set theory with no axion of coise (Shelah) .

A new paradigem in Set theory in now come even to small talkes.

Moshek

[Organic]

The numbers exist

What do you need to know that?

What if...?

"What if...?" is one of the most important questions that we ask when we develop something.

[moshek]

"Everything is a Number" , Pytagoras

And even the change of the Euclidian Paradigem:
Definition, axioms, theorems , etc

[matt grime]

What do you need to know that?

"What if...?" is one of the most important questions that we ask when we develop something.

Seeing as your question in you test is about counting beads it is implicit that you are presuming there is some method to count them. If there are no numbers, how do you count them? Irrespective of the difficulty in counting things and keeping track of which one is which.
Let me offer a different method of counting:

take the set of beads ina pile, pick one up, put it in your pocket and say 'one', then pcik another up and do the smae saying 'two' no difficulty there about people messnig them around or them al looking the same. The last things you say is the number of beads. Or are you asking about the philosophical issues about having a name for a number, and the concept of twoness existing independently of the words two, deux, zwei, dos,... metamathematics at most.

[Organic]

Matt,

One of the beautiful things that a man can do is a thinking experiment based on a "What if...?" question.

Most of the paradigms shift happened because people did not afraid to use this gifted ability to reexamine the obvious.

When you have eyes it is obvious that you can see, when you have memory it is obvious that you can count and define the concept of a NUMBER.

But "What if...?" questions can go beyond the obvious, and create an evolution in any asspect of life, including the fundamental concepts of Math language.

Numbers do not fully exist without our connition's abilities, and this is for real.


Seeing as your question in you test is about counting beads it is implicit that you are presuming there is some method to count them. If there are no numbers, how do you count them?

Numbers can exist iff there is an association between our memory and some internal(our own thoughts) or external elements.

[matt grime]

That is your philosophical position. A platonist would argue differently.

[moshek]

Matt:

I think that most of the mathematician in the planet of earth are Platonic.
They believe that when you discover something in mathematics is a discovery about some true, which is not in the real world. Few of them on not, take me as an example. And what about you?

Thank you
Moshek
[t)]

[matt grime]

Most of us are formalists. I think very few are platonicists. We have reasonable consensus on mathetical objects are, in the sense of what class of things are legitmately called mathematical, and as long as we are all honest in the use of logic, we make statements based on these hypotheses that we are in common agreement about, with the proviso that although we never explicitly deal with set theory unless we must, that there is the reassurance of having ZFC behind us that no one has shown to have any problems. And if they did then another person would come along and sort them out. I am perhaps a Wittgenstinian in remove owing to being taught by Gowers. An object is what it does is the paradigm there, for want of a better phrase. The natural numbers are the tools we use for counting things. We never explain it more than that, and few of us would feel the need to. If we had to we might offer some set theory.

Organic's Philosophy is somewhere between the two - although he seems to adopt a pragmatic approach to what the natural numbers are he often veers towards the idea that there is some platonic version that his is, and ours isn't, hence his statements about what properties they must have. Whereas my philosophy would state that we have words, one two three and so on and plus, etc that by common consent we have come to define in the current manner. They are useful for what they were designed for, and using them we can do a whole lot, including formally define all the objects Organic uses (it's a matter of what comes first, the name or the structure). However, when Organic uses the words one two three and so on, his isn't using them in accord with the convention of the rest of us. That ought to mean, in my opinion, that he should change the labels of his 'numbers' and use say, qone, qtwo, qthree, because they don't do the job that we have ascribed the other labels to. He would then need to define objects that are structureless quantities too.

Does it mean that the words are wrong or the usage of them is wrong?

I am happy for Organic to develop his philosophy. he seems to be doing a better job of it now, but I would still like him to properly describe the things he writes down.

Perhaps you should get together with him and talk about ostensive defintion and private languages?

[moshek]

Matt:

Thank you for your very kind explanation about your view on mathematics.
If you are formalist I want to ask you does mathematics is only a game with symbols?

Well Organic is Organic ( like me ) !

Yes you are right Organic numbers are defenetly not the conventional number

It will Be better if we write them like this:

1. 2. 3. 4. 5. 6. ...


Do you see the point?

And what is the definition of a point in the Euclidian mathematics ?

Thank you
Moshek
[t)]

[matt grime]

"does mathematics is only a game with symbols"?

I don't understand that.

Here might be an answer to the question that is usually asked in a manner like that.

Mathematics is not *only* a game with symbols. It is, as you like WIttgenstein, a language-game, perhaps. It is done in essence by the manipulation of symbols. Either on paper or in your head. How does one solve 2x=5? Divide by 5; x=2/5, but what do we mean by 2/5? It is that fraction that when multiplied by 5 yields 2. We don't need to explicitly state that as we've put it in a form that any reasonable person can agree with. Of course, I'm assuming the question was asked with the real numbers, or at least the rationals in mind. In mod 7 arithmetic the answer is of course x=6.

In Euclidean geometry a point is that which has neither length nor breadth.

What kind of point are you refering to?

[moshek]

Well you know that it is impossible to define a point but still all the geometry is base on construction with points !

Wittgenstein believes that maybe there mathematics with is base on the Geometry of the Klein bottle.

Here is the most difficult thing to understand in the whole story about
Mathematics:

Mathematics is not about discovery of things outside the world and also not about discovery of thing about the world

Mathematics is what mathemation are doing!


Can you see that point?

Moshek
[:)]

[matt grime]

Geometry has an axiomatic form. There are three axiomatized geometries, hyperbolic, euclidean and spherical. The Klein Bottle is a manifold and locally euclidean. I don't see what you're driving at. Mathematics might perhaps be what mathematicians do *mathematically*. Hence arbitrary labelling of diagrams without explanation is not mathematics unless one can describe it.

I don't think you have got the distinction between axiomatized geometry and its models.

[moshek]

The Erlagen program of Felix Klein was to analyze every Geometry by its symmetry Group. Now you can ask what is the fundamental symmetry of mathematics as a whole and not with 61 different fields. So you may got
a 4 dimension object which is the geometry interpastation to Goedel theorem.
we are part of this world and we need to develop a completely new mathematics and not Euclidian one. we must forgot all we know and just like young children to learn to count from the beginning..


But the result is mostly suprising
since you add finaly only one point to every concept
like organic is doing to numbers !

Best Regards
Moshek
[t)]

[moshek]

To my freind John:




One.

Two.

Three.

Organic..




Five

Six.

Seven.

Mathematics..




Only one point..

From the eternal..

And suddenly..

Everything is change..






Moshe Klein
:redface:

[Organic]

However, when Organic uses the words one two three and so on, his isn't using them in accord with the convention of the rest of us.


Dear Matt,

It is not quiet right, the convictional natural numbers are private cases of information forms in a mush more larger universe of ordered information forms.

When you use them, for example, to find that there are 67 different information forms in quantity 6, then quantity 67 does not give any information on the unique structure of each quantified form.

It means that it is not enough to say, for example, information form 13 (it is enough iff each information form has no-redundancy_AND_no-uanertainty information form) because we also have to explore its unique structure.

Therefore I give names like ET ( http://www.geocities.com/complementarytheory/ET.pdf )
or CR ( http://www.geocities.com/complementarytheory/CATheory.pdf ) to my information forms.

But new words or symbols are only one point, the other point is that I show a universe that can be systematically explored, developed and used by us to enrich Math language in more interesting information forms in infinitely many levels of information clarity degrees.

These ETs or CRs are based on a new kind of logic, which I call Complementary Logic
( http://www.geocities.com/complementarytheory/BFC.pdf ) where Boolean and Fuzzy Logics are private cases of it.

I also showed how Frege, the "father" of the Modern Logic, developed his logical system by using a private information form of my information forms ( http://www.geocities.com/complementarytheory/ConScript.pdf ).


And the last thing that I have to say in this post:

Everything which is exists (both abstract and non-abstract) can be changed, including the concept of the Natural Numbers.

[Hurkyl]

If you are formalist I want to ask you does mathematics is only a game with symbols?

I've always liked my answer to this suggestion:

Mathematics is only a game with symbols.
Science is the art of connecting those symbols to reality.

[Organic]

Hi Hurkyl,

And what is your motivation to play in this game of symbols?


Here's a question.

Can you prove something about quantity that cannot be proven through usual mathematical methods?

Here is a game with symbols that cannot be done by 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

[matt grime]

That requires you to explain what you mean by equals though, and all those sets, and to exlpain why 1+1+1+1+1+1 is not 5 when it is by definition of 1 and plus.l

[Organic]

Please open the new thread that I opened on this subject here:

http://www.physicsforums.com/showthread.php?p=170802#post170802

[moshek]

Matt :

I wrote :

mathematics is what mathematician doing.

And you correct that:

Mathematics might perhaps be what mathematicians do *mathematically*.

please explain me what is the different ?

Thank you
Moshek
:wink:

[moshek]

Hurkel :

Do you know way the "game" of mathematics
work in Science?

Thank you
Moshek
:wink:

[matt grime]

Matt :

I wrote :

mathematics is what mathematician doing.

And you correct that:

Mathematics might perhaps be what mathematicians do *mathematically*.

please explain me what is the different ?

Thank you
Moshek
:wink:

If a mathematician goes to the toilet, is the result mathematics? If a mathematician ignores the fact that certian number rings do not have unique factorization and "proves" fermat's last theorem, is it correct mathematics?

[moshek]

Dear Matt:

Thank you for your question to me about the last theorem of fermat and about the mathematitian who is going to the toilet!

I have share already with you more that 6 points in the History of the last 100 years of mathematics that may interpeted that we are standing infront of a tranjaction point in the history of mathematics. If you want i can repeat it again here and even add few mores.

If you really inside youself belive with this possibility even theoreticaly i can try to answer to you question about my infinity recursive definition to mathematics as a whole.

Please let me know first your aatitude to all this , so i can do my best to answer you.


Thank you
Moshek
:smile:

[matt grime]

tranjaction isn't a real word is it? also you don't want to pluarlize more to mores because that changes it's meaning to something quite strange, though not entirely unrelated. The point is that the question 'what is mathematics' is as thorny as any other philosophical question such as what is the nature of beauty. At least mathematics has a notion of correctness in that if you can defend you statements and show they follow from that which is reasonably assumed true. I don't think you've quite grasped all of the philosophical ideas you've mentioned, or their mathematical interpretation in terms of geometry, at least you cannot explain them clearly (geometry based on the Klein Bottle? An ambiguous statement given it's intrinsic nature as a manifold).

[NateTG]

In semi-response to Organic's post I thought I'd half take up one of his challenges:

Let S = NxT be the product of the natural numbers, N, with the set of all rooted finite trees (or directed graphs satisfying the obvious conditions), embedded in the plane, with the natural ordering on the branches/leaves.

I don't understand what you mean by natural ordering. There is depth-first and breath-first, ordering by branching number, or ordering by some sort of node index. It's unclear whether you're keeping track of the embedding as part of the "organic number". Can you clarify?


Let C be the subset of all possible { (n,t) | n in N, t a tree with exactly n edges}


Since you have some implicit embedding going on, I'm not sure whether this definition is sufficiently clear.


We define an operation + on elements of c:

(n,t)+(m,r) = (n+m,s) where s is the tree obtained by gluing the tree r onto the end of the first leaf.

Since this this operation is not closed over sets like 'C' (except for empty trees) - did you mean to use 'S'?


The subset (n,t) with t the trivial tree with 1 leaf and n edges, forms a copy of N under addition.

'trivial tree' is definitely not what you mean. A trivial tree would be a tree with the root node only.


We define * to be (n,t)*(m,s) by

(((...(((n,t)+(n,t))+(n,t))+...)+(n,t))

where there are m-1 addition signs.

So the tree structure of (m,s) is completey ignored?


again (n,t) with t the tree with n edges and 1 leaf, is a subset that forms a copy of N under multiplication.

It would help a whole lot if you were more careful with the notation.


neither + nor * are in general commutative, and I doubt they are associative either, but I can't be bothered to check, they are both well defined binary operations from CxC to C.



Now shall I claim that C is the new non-commutative natural numbers or not?

That's up to you, but so far C (or did you mean S) is not well-defined

[matt grime]

The ideas here are fairly well known. I didn't realize that trivial tree was reserved for only 1 edged tree, but the emphasis is on it being a tree with n edges that is trivial, or perhaps simple is a better word.

The idea of the ordering is that, if you'll allow me to take liberties,

|
|/

and its mirror image, which I won't attempt to draw, should be considered as different trees.

I was being deliberately vague, that tree would be an element of {(3,t)| t blah..}

and if I added it to itself I would have

|
|/
|
|/

C is closed under the addition operation take a tree with n edges, and a tree with m edges, glue the second's root on to the left most leaf of the first and you've got a tree with n+m edges. We could do this with directed graphs, which I would prefer, but this is supposed to be a mickey take of Organic.

This isn't supposed to make sense or be even the slightest bit relevant, as the completely stupid definition of a multiplication implies: yes it completely ignores the structure of the s-tree.

[Organic]

Matt,

Please draw the detailed tree forms of numbers 2, 3 and 4 by your system.

Thank you.

Organic

[matt grime]

I am not calling them numbers. Do you really need to see them they aren't very hard.

There are two elements of degree (I think that was the word I chose) 2

|
|

and \/

for degree three there are

|
|
|


|
|/

the mirror image of that in the vertical axis

\|/

for degree 4

|
|
|
|

|
|
|/

its mirror image and several more too tedious to draw out, get a pen and paper, you've got enough information to do it.

[NateTG]

So

|
\|

and

|
|/

are different, and you're simply enumerating the leaves in a clockwise fashion?

(So really you've got { (t,\preceq) } where t is a tree, and \preceq is an ordering of the leaf nodes of t with the additional properties that there is some node z (the tail) with z \preceq n \forall leaf nodes n \in t and for all a \preceq b \preceq c implies that the shortest path from a to b is no longer than the shortest path from a to c.)

These leaf nodes can then be readily enumerated using a floating base system.

I'm going to use some non-standard terminiology for a moment, since you've got a head (root), tail (least order leaf) and branches.

It seems like a more interesting (and natural) method for multiplication would be to replace each edge of one of the trees with a full copy of the other one attaching at the head and tail as appropriate. It would still work for the degenerate trees that you're using.

[Organic]

Matt,

1) there is no meaning to mirror images when we want to represent redundancy_AND_uncertainty variations.

If you don't think so, please show us how redundancy_AND_uncertainty are changed by a mirror image.

2) Please this time show the detailed trees of 2,3 and 4 without mirror images.

3) Please explain the relations between multiplication and addition in your system.

[matt grime]

For a bloody joke this is getting out of hand.

NateTG, if you want to define other operations please do so, I don't care in the slightest, but I was attempting to get the silliest defintions I could.

Organic, why would I require uncertainty and redundancy to change under mirror images? I don't in my system, which I won't justify becuase I don't have to.

The realtiohsip between + and * is given to you in the definition of * as repeated +.

No, I can'be be bothered to draw out the diagrams, why should I? I don't actually think this idea is useful or interesting to anyone but you. It is a silly and pointless exercise I cooked up in a couple of minutes. I'm sure if I felt like it I could construct a system doing exactly what you require, but as you never explain what you require clearly it would be a tiresome exercise - after all, why are you now saying mirror images must alter redundancy and uncertainty? I don't remember you saying that ever before (you didn't mention mirror images) so how do I know you won't change your mind again?

[Organic]

Matt,

By using "\" "|" and "/" symbols I can represent 1,2,3 and 4 in this way:

|


|
\/ |


|
| |
\|/ |/ |

|
|
| | || | |
\\// \/// |// \/\/ |\/ || \|// |// |

There are no left-right switches or mirroring changes in my system.

Now, please use the same three basic symbols to represent your trees, without left-right switches or mirroring changes.

Here is an example of left-right switches:

\/|| |\/|

Here is an example of mirroring changes:

| |
|/ \|

[matt grime]

fine, delcare all mirror image trees to be identified, i don't really care. most of the things you drew aren't trees cos they don't have a root, but that could jst be the restrictions of html coding, and you've repeated lots of them too.

[Organic]

Matt,

If you can't draw your detailed trees, we can understand that you have no method to define them.

So, you did not succeed to produce any system where multiplication and addition are complementary operations.

and you've repeated lots of them too.

Please give an example

[matt grime]

No, I can't be bothered to draw them. If you cannot draw all trees with 4 edges and so on then that's your problem. As you have not said what it means for addition and multiplication to be complementary the second assertion is a little moot, isn't it?

To draw a tree/directed graph. Pick a base point. Draw arrows out of the base point, from the tip of each arrow draw more arrows going out that do not touch any other arrows. Repeat.

the elements of degree n are all the diagrams you can draw with n arrows, I require you to order the edges leaving a node so that you can differentiate between certain trees. You've declared that you ought not to do that, but why? I disagree, and you cannot prove I'm wrong.

It is not clear what

\\// \/\/ \||/

etc that you drew are, seeing as they all have depth 1 and should all thus be the same tree - where is the root point?

You do know that a tree is a (directed) graph without loops?

[Organic]

This is my system:


0
|

1


11
00 1|
\/ 0|

1 2


222
111 11 2|
000 002 1|
\|/ \// 0|

1 2 3


3333 33 33
2222 22 22 222 3|
1111 1111 11 1111 11 111 11 2|
0000 0000 1|00 0000 1|00 1||1 0003 0023 1|
\\// \/// 0|// \/\/ 0|\/ 0||0 \|// \/// 0|

1 2 3 4 5 6 7 8 9

and it cannot be represented by "\" , "/" and "|" notations (as you choosed to do) because, for example, in a given quantity 4, tree-2 has the same shape as tree-8.

So it is not so trivial as you think.

Now, please find an accurate way to draw your trees, and please explain what data you give to each branch, as I did.

[matt grime]

I am assigning no data to my diagrams; I didn't say yours was trivial (it isn't very interesting, but that's different); I have noi desire to draw out these things, they aren't interesting, useful or of any point what so ever, just like yours. You said there was nothing maths LIKE your theory. There is. As you've never bothered to explain your theory accurately it's only a reasonabl approximation, the best we can do when there is incomplete information.

[Organic]

(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)

All left-right variations are ignored, please see the examples below:

{1,1,1,1}

{{1,1},1,1} (left-right {1,{1,1},1} and {1,1,{1,1}} are ignored)

{{{1},1},1,1} (left-right {1,{{1},1},1} and {1,1,{{1},1} and {1,{1,{1}},1} and {1,1,{1,{1}}} are ignored)

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

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

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

{{1,1,1},1}

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

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

The first one is a multi set {1,1,1,1}

The last one is a normal set {{{{1},1},1},1}

All other collections between them are the ordered collections of hybrid sets.

Can you use this to show your system?

[pallidin]

Please excuse my interjections, as I have only briefly followed this thread and the related one from the Math section.
Organic, I am a little confused as to the thrust of your arguments due to all the rambling. Could you re-state them?

[Organic]

Hi pallidin,

Multiplication and addition are complementary operations, which mean that each operation simultaneously defines and prevents the other operation.

"Pure" multiplication can be operated only among identical objects, where "pure" addition can be operated among unique objects.


1
(+1) = {1}

2
(1*2) = {1,1}
((+1)+1) = {{1},1}

3
(1*3) = {1,1,1}
((1*2)+1) = {{1,1},1}
(((+1)+1)+1) = {{{1},1},1}

4
(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)
5
...


Please show what is not understood in the above example.

[pallidin]

OK.

Well, I understand what you have shown, but not sure of the point.

[Organic]

Please look at my website:

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

[moshek]

Matt

How you write correctly in english "tranjuction " point.
I mean by this a "the point of the big change".

thank you
Moshek

[cookiemonster]

I think I'm going to let myself be dragged into this for just about exactly one post...

Organic, do you have a concrete example of the use of your system? I see lots of you arguing with matt about how he's wrong and you're right and he's obtuse and you're not, etc., etc. And I see lots of little diagrams which have no meaning to me, which is probably my own fault. And I see lots of claims that this may or may not have use to physical theories that require some uncertainty.

I'll be honest with you. I really haven't cared a bit about your theory because I haven't seen its use. To me, math isn't a whole lot more than a tool or a game. It'd probably take me a lot of work to go through and try to understand what you're getting at (no offense, but your explanations seem to be exactly the same, which means it won't help the second time if it didn't the first), and unless it's worthwhile, I don't want to waste my time.

So can you show me it's worthwhile? Can you show an example? Can you throw your numbers at the classical expression for U + T = E and have Schrodinger's Equation appear? Or is this an unfair comparison between your numbers and the currently established numbers? If so, is there a fair comparison?

cookiemonster

[Organic]

Hi cookiemonster

I hope that you will find the cookies that you like here:

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

My theory of numbers is also new tools for new games.

If you understand their properties, then you can find by yourself how to play with them.


I see lots of you arguing with matt about how he's wrong and you're right and he's obtuse and you're not, etc., etc.

I don't say that Matt is wrong (he says this about me) all what I say is that because he ignores the structural/quantitative relations as fundamental point of view of the NUMBER concept, he cannot understand my number-system.

[cookiemonster]

Your link doesn't mean anything to me. All I see are the same diagrams I've been seeing for the past million posts, which still don't mean anything to me.

"If you understand their properties, then you can find by yourself how to play with them."

Yes, if I understand them. But I don't see any reason to put in the work to understand them. Why should I understand them? Where's Schrodinger's Equation? Where's the results?

cookiemonster

[moshek]

To Organic
with his new numbers system:


----------------------------------------------


For Mathematics

The current big-band
His here real Glory.
Milky way is around us.
A solar system was created.

Everything is a number
Said Pythagoras
While he could hear
The music of the spheres.

But so many water
Cover the head of Hipasus
After he discovery
The secret of irrationality.


Maybe Euclids hide the story
For the protecting the axiom
Of the parallels
To establish his own mathematics.

While Newton calculate
The end of the world
Leibniz with the monads believed
A unify language must exist.

Goethe could see here
With the generic type
But he just did not
like or know mathematics

Hilbert was staying
So misunderstood
With his list of 23 problems
and the organic unity.


A.Connes with
Noncomutativs geometry
100 to Hilbert end with
some new understanding.

M.Athiya for his Index
And K theory
Talk about here
As some Enigma.

I Stuart with his vision
Share her flexibility
In his Epilog
The nature of numbers

Wittgenstein say
We should be Aliens
To see here in
The bottle of Klein.


From the top mountain
Of the Rieman hypothesis
We can see the real mount Analog
And Hear its’ sixth symphony. ’


Einstein did a real
First step of a child
When he ask how we
measure a length.

Only if we could See again
The world Like children
We will count again
Now from the beginning 1. 2. 3.




Moshe Klein 4.4.04

[Lonewolf]

Ok, how do you obtain the rationals, irrationals etc. out of this theory?

[moshek]

Only by Analogy no definition:

Since the relation between irrational number to rational numbers
Is like "Organic mathematic" with here new center of the organic unity of mathematics to the Euclidian mathematics where the center there was logic.

[Lonewolf]

I do beg your pardon?

[moshek]

For what Lonewolf ?

[Lonewolf]

"Organic mathematics", and an explanation of "Organic unity" is required before I can understand what the centre of it is.

[moshek]

Ok !

what do you understand from this qoute of Hilbert :


The organic unity of mathematics is inherent in the nature of this science, for mathematics is the foundation of all exact knowledge of natural phenomena. That it may completely fulfil this high mission, may the new century bring it gifted masters and many zealous and enthusiastic disciples!

[Organic]

Hi Lonewolf,

Ok, how do you obtain the rationals, irrationals etc. out of this theory?

please read:
http://www.geocities.com/complementarytheory/NewDiagonalView.pdf

and this:

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

More can be found here:

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

[Organic]

Your link doesn't mean anything to me. All I see are the same diagrams I've been seeing for the past million posts, which still don't mean anything to me.

"If you understand their properties, then you can find by yourself how to play with them."

Yes, if I understand them. But I don't see any reason to put in the work to understand them. Why should I understand them? Where's Schrodinger's Equation? Where's the results?

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

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

My new nutural numbers are like wavicles.

Therefore if you use them, you get a natural picture of QM elements.

[matt grime]

Exactly in what context did I say you were wrong, organic? You are wrong about many things, but then you freely admit to not knowing much maths. You've not yet offered any reason as to why the strucuture of the natural numbers should be part of their definition - the axioms of a group (finite) do not say that the order of an element must divide the order of the group but that is part of the structure of group.

You have offered in the last few posts the first defintion of complementary pertaining to multiplication, but it didn't make much sense.

You might also care to explain why your definition of what the natural numbers ought to be doesn't agree with the usual definition. Some might consider that to be a problem. Not you.

So from first prinicples, why don't you demonstrate how, using the set of organic numbers, O, say that you obtain a meaningful system? You might be able to do so. Show how they can be used to solve an equation, model the flow of water round a sphere, be localized to form a field... Or even explain what they are used for. (there are many kinds of numbers organic, such as p-adic... any thoughts about that?)

[Organic]

Matt,

Multiplication and addition are complementary operations, which mean that each operation simultaneously defines and prevents the other operation.

"Pure" multiplication can be operated only among identical objects, where "pure" addition can be operated among unique objects.


1
(+1) = {1}

2
(1*2) = {1,1}
((+1)+1) = {{1},1}

3
(1*3) = {1,1,1}
((1*2)+1) = {{1,1},1}
(((+1)+1)+1) = {{{1},1},1}

4
(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)
5
...


If you understand this ordered relations between a multiset and a "normal" set, then you can see that these ordered elements can be used to construct an ordered table of infinitely many information forms, that can be used as fundamental building-blocks, that when connected to each other, can help us to research much more complex models than the conventional number system.

Please this time try to read all what I wrote here, and see for yourself a gate to complexity:

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

If this time you understand what I am talking about then try to connect this point of view to your knowledge and by this connection please tell me what do you find.

Thank you

Organic

[Hurkyl]

Why is there no ((1*2)*2) or ((1*2)*(1+1)) or ((1*(1+1))*(1+1)), et cetera?

[Organic]

In my system multiplication can be operated only between identical elements,
where id is both structural and quantitative.

For example:

((1*2)*2) = ((1*2)+(1*2)) = {{1,1},{1,1}}

((1*2)*(1+1)) does no exist because the internal information structure of (1*2) and (1+1) is different, for example:

(1*2) = {1,1}

(1+1) = {{1},1}

Shortly speaking multiplication cannot be operated between elements, which are not equal by their structural properties.

Please look again at page 7 (in the paper, not in the acrobat screen):

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

[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:

1
(+1) = {1}

2
(1*2) = {1,1}
((+1)+1) = {{1},1}

3
(1*3) = {1,1,1}
((1*2)+1) = {{1,1},1}
(((+1)+1)+1) = {{{1},1},1}

4
(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)
5
...

[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.

[url]
My new nutural numbers are like wavicles.

Therefore if you use them, you get a natural picture of QM elements.

You've asserted this before. I want to see it. Can you describe for me the Hydrogen 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:

1
(+1) = {1}

2
(1*2) = {1,1}
((+1)+1) = {{1},1}

3
(1*3) = {1,1,1}
((1*2)+1) = {{1,1},1}
(((+1)+1)+1) = {{{1},1},1}

4
(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)
5
...

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.

A general graphic description of a CR

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

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

1
(+1) = {1}

2
(1*2) = {1,1}
((+1)+1) = {{1},1}

3
(1*3) = {1,1,1}
((1*2)+1) = {{1,1},1}
(((+1)+1)+1) = {{{1},1},1}

4
(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)
5
...

[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:

0
.
1 = |
*


1 1
0 0 0 1
. . . .
| | | |
2 = |___|_ |___|
| | *


2 2 2
1 1 1 1 1
0 0 0 0 0 2 0 1 2
. . . . . . . . .
| | | | | | | | |
3 = | | | |___|_ | |___| |
| | | | | | |
|___|___|_ |_______| |_______|
| | | *


-------------->>>
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 0 1 2 3
. . . . . . . . . . . . . . . .
| | | | | | | | | | | | | | | |
|__| |__| | | | | |__|_ | | |__| | |
| | | | | | | | | | | |
| | |__|__|_ | |_____| | |_____| |
| | | | | | | |
|_____|____ |________| |________| |________|
| | | | *


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:

1
(+1) = {1}

2
(1*2) = {1,1}
((+1)+1) = {{1},1}

3
(1*3) = {1,1,1}
((1*2)+1) = {{1,1},1}
(((+1)+1)+1) = {{{1},1},1}

4
(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)
5
...