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

[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

 

[moshek]

Hurkel :

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

Thank you
Moshek

 

[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

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 believe 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

 

[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:

[b]
0
|
[/b]
1
[b]

11
00   1|
\/   0|
[/b]
1     2

[b]
222
111   11    2|
000   002   1|
\|/   \//   0|
[/b]
 1     2     3

[b]
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| 
[/b]
  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.

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

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

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

[b]4[/b]
(1*4)               = {1,1,1,1} <------------- Maximum symmetry-degree, 
((1*2)+1*2)         = {{1,1},1,1}              Minimum information's 
(((+1)+1)+1*2)      = {{{1},1},1,1}            clarity-degree
((1*2)+(1*2))       = {{1,1},{1,1}}            (no uniqueness) 
(((+1)+1)+(1*2))    = {{{1},1},{1,1}}
(((+1)+1)+((+1)+1)) = {{{1},1},{{1},1}}
((1*3)+1)           = {{1,1,1},1}
(((1*2)+1)+1)       = {{{1,1},1},1}
((((+1)+1)+1)+1)    = {{{{1},1},1},1} <------ Minimum symmetry-degree,
                                              Maximum information's  
                                              clarity-degree                                            
                                              (uniqueness)
[b]5[/b]
...

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.