A new point of view on Cantor's diagonalization arguments

[Organic]

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.

 

[matt grime]

Originally posted by moshek
Matt :

Did you ever read the 4 last lines
of Hilbert lecture at Paris 1900?


Organic already share with us that he dedicate almost 20 years from his life in trying the develop new perspective about mathematics and he share with us his discoveries very gently.

That fact that he is not famiyar with ordinary mathematics ( he admit that already ) is not necessarily relevant to the question if him material have the potential to make an Organic sift in mathematics.

Take care
Moshek

That'd be the Hilbert who inspired Organic's username? The one who also said we need never stray from the paradise Cantor created? Wonder why that hasn't sunk in?

You posted saying organic might be interested in some mathematics conference. What evidence do you have that he would even understand the simplest talk there, or care? I remember offering him a link to some papers of Keller explaining operads and A infinity algebras which give structures on trees that he said mathematics cannot describe. He dismissed them as worthless in less than 20 minutes.

 

[moshek]

Matt:

I am sorry to hear that Organic dismissed significant mathematical paper. Maybe because he have not background to read it.
Well at list you can see that he was influence from Hilbert in the end of his famous lecture since he is trying to develop Organic type of a number with the duality of Quantity and structure.

As far as I can understand big change always started from to most simple thing.

Do you familiar with Wittgenstein attitude to mathematics?


Best
Moshek

 

[Organic]

Matt,

Let me refresh your memory:

I wrote:

Matt, can you give me some address where i can read about A-infinity algebras?

Thank you.

Organic

Last edited by Organic on 01-22-2004 at 01:50 PM

Your response was:

And,no *I* won't tell you anything about A-infinity algebras. I don't beleve that will accomplish anything as you've yet to understand the basics of common mathematics, and these are very advanced concepts.

Then I have found myself these articles, and then I wrote to you:

Matt,

If you wrote this:

Your multiplication is not well defined then, because your 'numbers' are not well defined, that is there is no single element that represents it uniquely. For instance you give me several different answers for 2*3. To deduce anything from this you must acknowledge that '3' is not a number in your system and to call them numbers is misleading.

Then I have to say that you are much more closed system then I thought, that has 0 ability to understand new fundamental ideas about what numbers are.

You are in your closed system cannot understand how multiplication and addition are complement operations that do not changing the quantity of the number, but only its internal scructure, by breaking and unbreaking its internal self symmetry.

Now, I see you jump and saying: "here comes more undefined terms".

All the rigoruos terms are infront of your eyes but for you terms is written text, not for me.

Also I looked at:

http://arxiv.org/PS_cache/math/pdf/9910/9910179.pdf

http://arxiv.org/PS_cache/math/pdf/0108/0108027.pdf


They do not based on my fundamental ideas, therefore cannot express my mathematical system.

 

[matt grime]

My apologies for my poor recollection. But you did dismiss out of hand these objects without thinking about them for anywhere near long enough to even begin to understand them.


I am familiar with Wittgenstein, and Gowers, who is currently the most vocal proponent of this position on mathematics.

This alleged multiplication on numbers. You do realize that it doesn't even take a pair of integers and then produce a third, which we can call their product. So to say it is the mulitplication that is 'complementary' to addition is disingenuous.


It is an operation on (certain) rooted trees, some subsets of which you label with elements of N. If you cannot even begin to see why you ought to learn about operads given this then it is you who is closed off to 'new' ideas. You've almost never accepted anyone else might have a valid point of view on this.

 

[moshek]

Dear Matt !

Please sent me more information about Gowers work.

I am really glad that you know already Wittgenstein attitude to the possibility of creation new mathematic and the place of Klein bottle in this creation by adding one more dimension to mathematics as it appear in today paradigm which base on Logic in it center !

Have you notice already on the similarities of Organic work in mathematics to Prague work on creating first order logic?

Thank you
Moshek

 

[matt grime]

He's got a fields medal, I think you can locate him withtout too much trouble.

It is hard to notice any similarities with what organic writes and extant maths because he misuses terms all the time. My favourite at the moment being the insistence that the infinite rooted binary tree is a Cantor Set.
His abuse of the term enumerable or countable is now just annoying, as is his refusal to learn about any of the words he uses.

His use of trees is reminiscent of operads, and the operations are those of semi-groupoids or monads. He often asks us to offer current mathematical objects that does what his do, we have, with these, yet I doubt he's ever read about these to see. Nor does he state what his things 'do' (cf my repeated requests to find out how one decides which trees exhibit 'uncertainty') which as a Wittgenstein-follower you should admit is the most important thing.

 

[Organic]

Matt,

This alleged multiplication on numbers. You do realize that it doesn't even take a pair of integers and then produce a third, which we can call their product. So to say it is the mulitplication that is 'complementary' to addition is disingenuous.

You can't understand it do you?

Any product in my number system is a structural/quantitative product.

Therefore multiplication and addition must be examined through this point of view, and from this point of view multiplication and addition are complementary operations.

This is a new approach, and partial treatment about it maybe can be found in A-oo algebras here:
http://arxiv.org/PS_cache/math/pdf/0108/0108027.pdf
in what is called "multiplication inside the inner-product-diagram" (page 26).

 

[moshek]

Matt:

Wittgenstaein said : "..Mathematics have no foundation in Set theory or in any other theory. Mathematics is depend only on our living.." .

maybe this is way Organic ask us what is the real meening of the 3 points in :

{1,2,3,... }

Or about the very similarity of Rashel paradox and Cantor diagolazation metod to prove that R in uncontable.

I found something about Gowers , thank you !
I understand that his new idea about mathematics deal with the connection between mathematics and Physics.

Well i defiantly agree with that direction.

Do you know the Michel Atiya ( Also a well known fields medal) said in his lecture at the conference about the unity of mathematics at Harvard university ( I was there)
that we are waiting to a new Newton that will broke the Enigma between mathematics and Physics

So do you think that maybe Gowers
is our new Newton ?

Thank you
Moshek

 

[matt grime]

I do get it, Organic, and because I get it I see that you're being imprecise:

you are calling it a multiplication on N, aren't you?

take two elements in N, say [2] and [3], you have shown what [2]*[3] isn't another natural number, [n].

[2] and [3] in your model are a sets of trees satifying certain properties, their product give you some of the trees in the set [6], doesn't it?

I don't think you can then be justified in calling it a multiplication on N.

I am a pure mathematician, and am happy with abstraction and objects that are apparently meaningless, but you do not provide enough information to say what's going on! Repeatedly I've said you're more than welcome to develop all this machinery, it's just that no one else understands what you're doing because you do not offer enough explanation or define anything in terms that allow other people to reconstruct your objects. If you forgot all the mumbo-jumbo about maths needing to remember the people participating in it, you'de get a lot further. Of course you can't do that because to you it's vitally important, though no one else can see why.

 

[moshek]

Hurkel !

I ask you about the futher
so way did you answer me
about the past ?

Moshek

 

[matt grime]

Originally posted by moshek
Matt:

Wittgenstaein said : "..Mathematics have no foundation in Set theory or in any other theory. Mathematics is depend only on our living.." .

maybe this is way Organic ask us what is the real meening of the 3 points in :

{1,2,3,... }

Or about the very similarity of Rashel paradox and Cantor diagolazation metod to prove that R in uncontable.

I found something about Gowers , thank you !
I understand that his new idea about mathematics deal with the connection between mathematics and Physics.

Well i defiantly agree with that direction.

Do you know the Michel Atiya ( Also a well known fields medal) said in his lecture at the conference about the unity of mathematics at Harvard university ( I was there)
that we are waiting to a new Newton that will broke the Enigma between mathematics and Physics

So do you think that maybe Gowers
is our new Newton ?

Thank you
Moshek

No.

Gowers prefers to deal with elementarily stated problems, that are therefore hard to solve.

If you want a mathematical physicist then Baez, Dolan, Atiyah (still. Mathematically I am his partial grandson as I was supervised in my phD for a year by one of his students), Witten, Grothendieck (if he's stoppped doing biology), Orlov or Lusztig seem more likely.

Perhaps Hurkyl doesn't desal with the futher because there's no such word as futher. Do you mean future or father?Mathematics is precisely what we make it, I agree, which seems to conflict with Organic's apparently previously held belief that the natural numbers do not exist independently of set theory.

 

[matt grime]

Originally posted by Organic
Matt,

[2]*[3]=[6] but because my nanural number is a structural/quantitative product, there is more than one solution to [6], which are dependend in the inner structure of [2] and [3], for example:

[2]=(1,1) , ((1),1)
[3]= (1,1,1) , ((1,1),1) , (((1),1),1)

Therefore:

[2]*3=((1,1),(1,1),(1,1)) or (((1),1),((1),1),((1),1))
[3]*2=((1,1,1),(1,1,1)) or (((1,1),1),((1,1),1)) or ((((1),1),1),(((1),1),1))

which justifies my assertion that the is no unique element that represents [n] and the multiplication is therefore not well defined - something you riduculed in that reposted message.

 

[moshek]

Matt :

Do you know that Alein Connes the great mathematician who invent Non -commutate Geometry ( also a fields medal) end with his interesting lecture the conference "100 to Hilbert" at August 2000 at U.C.L.A when he said that we need today a new understanding in mathematics that is base on Geometry and not on logic ?

Organic:

Thank you for sharing with us
your very interesting noncomutative arithmetic.


Moshek



Who was killed by a solders and said:

Bring me one point so i can change the whole world .

 

[moshek]

Matt :

Good to know that !

I am Sorry for my English mistake in my question to Hurkel
yes you are right i mean future. ( of mathematics)

Well your Grandfather gave the best and maybe the only lecture that was relate really to the question of the unity of mathematics ( as was posed by Hilbert at the end of his lecture).

Please ask him if i quote him correctly about a change in the paradigm of mathematics !.

And if he said also :

"We are waiting today to a new Newton ?

Maybe you know another possibility if it is not Gowers ?

Thank you
Moshek

 

[Organic]

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


Matt,

Natural number system of Standard Math is not simple but trivial, and my attitude is to give it the power of simplicity instead of the weakness of triviality.


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

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 mean 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

 

[moshek]

Dear Organic:

We all know already that you are not so good in what is called mathematics...


But :

Your gentle treatment of number with there inherent duality is seeing the world in the eyes of a child.

Like Einstein did to relativity theory it is the only way to create a unmoors change with the paradigm.

So you share with us by this
a positive intepatatino to Gödel theorem.

My Best wishes
Moshek

 

[matt grime]

The quantum shift is mathematical: quantum groups, quantum cohomology, quantized universal enveloping algebras, quantum chaos, even quantum mechanics (I learned that in a maths degree...), I've even been learning about quantum linear algebra. And then there's quantum computing and quantum information theory, both well established and mathematical. And did I mention quantum Kac-Moody Lie algebras? Deformation Theory? q-schur algebras, quantum binomial coefficients?

 

[moshek]

Matt:

well It's all about modeling of the world, like Newton did !
Einstein put us in the world but still use Newton mathematics.

Did you heard or read Wolfram Book "new kind of science"


Please Ask Michel Atiya about redefinition of the 6 problem of Hilbert ! maybe Organic is trying to share with us his solution to this very difficult problem.

as you know the 6,8,16 problems
are still opme in the list.

Thank you
Moshek





p.s I study now his very nice book on commutative algebra of Atiya and Mecdonalds.

 

[Organic]

Matt,

Instead of reading the details of my previous post you take some name and play with it.

Please show me some influence on basic thing like the natural number, by one of these theories.

 

[moshek]

Dear Matt:

We can say that the origin of the possibility to a paradigm change in mathematic was at the end of Hilbert famous lecture. Unfortanly that math world relate mostly to the famous 23 list of open problem which Hilbert himself said that the are only example.
This was 5 years before Einstein appears to the world. Wittgenstaein pose already about this possibility also when he talk about develop anew dimension to mathematics in the geometry of the Klein bottle. In the Epilog of the nice book of Ian Stewat “Nature Numbers” he also talk about he need to invent new –mathematic which will be flexible and rigid together. Alein Connes talk about a new understanding that may come to mathematic beside the regular logic. And Michel Atiya lecture about the need to broke the enigma of the connection between mathematics and physics

“We are looking for a new Newton today..” he said at the conference about the unity of mathematics in Honored of Israel Gelfand 90 birthday.

Yes Hurkal is right !

A similar change happened already but it was ~2,500 years ago when Hipasus of metapontum discover that irrational number exist. Which Contradict Pythagoras “Every thing is Number”.

I ma sure you know the legend what his teacher did to him after that.

Organic share with us here his discovery about Number when time is Vanish.
So you must hold the quantity and the structure always together.



Take care
Moshek

 

[matt grime]

Organic would need to demonstrate the the natural numbers are not sufficient for the job they are used for for that parallel to hold here.

The greeks thought that all numbers on their number line were rational originally. This is not true, therefore their assumption is wrong.

Where do we go wrong with the definition that the natural numbers are the basic counting blocks of finite sets? What has gone wrong?

Sure we can add all the extra structure to it you want, but it doesn't alter the basic facts.

You could argue the 'inventing' the negative integers, then the rationals, then the reals and then the complexes are all paradigm shifts, but they don't mean the naturals are incomplete within themselves.

We dont' assume those are all the arithmetical objects that exist.

As I keep telling organic he's just playing around with basic operads and monads, if he wants to add extra structure to them that's fine, 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. That is not the objection to what he's doing. The objection is that he's making wild unsubstantiated and plainly false claims about things he doesn't understand.

 

[moshek]

Matt:

You Ignore by your question to Organic
all my background that i share with you here
as i promise you few hour ago!

Moshek


I will let Organic to answer your question about his numbers.

 

[Organic]

Matt,

First, you have to show me that you understand what I am doing.

For example, it is the second time that you write general things about my work instead of reading and and reply detailed comments on this:

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


Matt,

Natural number system of Standard Math is not simple but trivial, and my attitude is to give it the power of simplicity instead of the weakness of triviality.


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

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 mean 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


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

The quantum shift is mathematical: quantum groups, quantum cohomology, quantized universal enveloping algebras, quantum chaos, even quantum mechanics (I learned that in a maths degree...), I've even been learning about quantum linear algebra. And then there's quantum computing and quantum information theory, both well established and mathematical. And did I mention quantum Kac-Moody Lie algebras? Deformation Theory? q-schur algebras, quantum binomial coefficients?

Please show me some influence on basic thing like the natural number, by one of these theories.

 

[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 don't understand that the stantard 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

 

[matt grime]

Originally posted by Organic 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.

This would only hold true if I had ever said that the Natural Numbers are the only thing in mathematics. We have N, simply, N, the set of natural numbers. Then there is (N,+) which is a semi-ring. Then the is (N,<) the ordinals. And who can forget (N,+,<) the ordered semi-ring. On top of that one might do (N[-1],+), ie Z under adding formal inverses. Not forgettiing the ordering too. What about localizing and getting a field, Q, and completing to get R, and closing to get C? Then there's the division rings and Wedderburns' structure theorem over that.

Of course there is the general theory of semi-rings etc that 'has N as a private case of a larger language', in you words.

There is a whol rich tapestry of objects that have N, Z, Q or R as the simplest version. This apparent paradigm shift is already there.

If you incidentally knew about the quantized structures above and the rigorous framework behind quantum mechanics, then perhaps you'd understand the analogies.

 

[Organic]

Matt,
Two critical things you ignore when you define N members:

1) A research of your ability to count:
http://www.geocities.com/complementarytheory/count.pdf

2) that natural numbers are first of all information forms, therefore
Their minimal existence must start form here:

(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)

After these two "MUST HAVE" steps, we can continue to develop the next numbers in the number system.

 

[matt grime]

Why are these must have steps? We can define the natural numbers without them, so they obviously aren't must have are they? Or are you forgetting the DEFINITION of the natural numbers (as a set, just a set, with no other structure)?

 

[Organic]

Matt,

The answer is very simple.

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.

Or are you forgetting the DEFINITION of the natural numbers (as a set, just a set, with no other structure)?

A set is only a framework that helps us to explore our ideas, no less no more.

When there is a paradigm shift this framework is chaneged too.

 

[matt grime]

It would be nice if you told us what the uncertainty and redundancy of a number is. So that we know what we MUST be aware of.

 

[Organic]

The answer is here:

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

 

[matt grime]

Originally posted by Organic
The answer is here:

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

Ah, of course! The answer to my question is in another pdf! This one doesn't even mention the words uncertainty and redundancy. In what way is that an answer?

Do you mean to imply that numbers are uncertain because if we have set of identical objects we cannot distinguish between them? I don't see why that makes numbers uncertain. I see why it makes identifying identical objects impossible, but that has nothing to do with quantity. Suppose I just had one bead. I blink, I see an identical bead in the same place. is it the same bead? I do that for more than 1 bead.. what has quantity got to do with it?

 

[Organic]

From your response it looks that you did not read the pdf file.

Please reply if you have some technical problems to open the pdf.

Thank you,

Organic

 

[matt grime]

I opened the pdf link you gave and didn't see the word uncertainty mentioned once, please tell me which line it is on. Or, as the pdf is only a page long, try stating in plain simple English what you mean the the uncertainty of a number, or its redundancy. It's another simple request. Actually it's the same simple request isn't it? A simple paragraph starting:

The redundancy of a number is...

or perhaps it ought to start

A number is redundant if...or even,

given a number n, and some object related to n, then the object is redundant if...

 

[Organic]

Dear Matt,

You are going too far.

This pdf is not a technical paper, but a simple test that answer to the question:

what are the minimal conditions that give us the ability to count?

Please read it again from this point of view, and don't search for any definitions there, just try to understand this simple test.

Thank you.

Organic