Exploring Non-Commutative Natural Numbers

[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.

 

[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

 

[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

 

[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 referring 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

 

[moshek]

To my friend John:

One.

Two.

Three.

Organic..




Five

Six.

Seven.

Mathematics..




Only one point..

From the eternal..

And suddenly..

Everything is change..






Moshe Klein​

 

[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:

https://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

 

[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.