Organic Numbers

1. Mar 23, 2004

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?

2. Mar 23, 2004

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

3. Mar 23, 2004

moshek

Dear Matt:

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

Best
Moshek

4. Mar 23, 2004

matt grime

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.

5. Mar 23, 2004

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.

6. Mar 23, 2004

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.

7. Mar 23, 2004

Organic

Matt Grime,
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).
You right because you don't know what you are doing.
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 â€œquantitative-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

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

Last edited: Mar 26, 2004
8. Mar 23, 2004

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.

9. Mar 23, 2004

Organic

1)Redundancy AND uncertainty

2)Show now where is the tree?
Code (Text):

(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

10. Mar 23, 2004

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.

11. Mar 24, 2004

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.

Last edited: Mar 24, 2004
12. Mar 24, 2004

matt grime

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,

13. Mar 24, 2004

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.

Last edited: Mar 24, 2004
14. Mar 24, 2004

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.

15. Mar 24, 2004

Organic

16. Mar 24, 2004

matt grime

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?

17. Mar 24, 2004

Organic

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

18. Mar 24, 2004

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?

19. Mar 24, 2004

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.

Last edited: Mar 24, 2004
20. Mar 24, 2004

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.

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.

21. Mar 24, 2004

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.

Last edited: Mar 26, 2004
22. Mar 24, 2004

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.

23. Mar 24, 2004

Organic

Matt,

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

24. Mar 24, 2004

Organic

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...

25. Mar 24, 2004

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.