A new point of view on Cantor's diagonalization arguments

[Hurkyl]

The problem, as I see it, is that you seem to be spending a lot of effort to assert that your tools talk about things that they don't.

For instance, what ever concept of "quantity" you have in your system, it is different from what a mathematician calls cardinality. You've done a great deal of asserting that your tools are proving that things mathematicians know about cardinality is wrong, when all you're really doing is discovering the differences between cardinality and your concept.

You seem almost fanatical in this pursuit, which is why many people get turned off by your theorizing. To be frank, because of your approach, I lost all interest in your ideas because you just couldn't seem to get past the "Look, I've proved mathematics wrong!" mentality.

The main reason I keep posting in your threads is because I think you're not a hopelessly lost cause like most so-called crackpots. (and that I'm a glutton for punishment!) You still seem to have the "This is so obvious, why can't they see it the way I do?" mentality about things, but you do sometimes seem to learn and adapt in the face of criticism.


Also, I hope you don't think that rigor is all mathematicians do. It is the ultimate standard, but intuition and heuristics have a lot to do with how we do things. For instance, on a great many problems I will start with "this is how my gut says to do this problem", and then if it looks promising, I begin to fill in the details. Filling in the details often gives me a good proof, and other times it illuminates a flaw I made in my intuitive reasoning. There is a lot of mathematics built up on things that are still conjecture; for instance, there are a lot of theorems of the form "If the Riemann hypothesis is true, then this other statement is true". Mathematically, one does not need to have proved the hypothesis with full rigor to reason about things; you just have to acknowledge that you haven't derived an "absolute" fact, but instead a "relative" fact.

 

[phoenixthoth]

hmm...

 

[matt grime]

Oh, well, time to feed my habit again.

It appears by 'complete' you mean a set is complete iff you can write down every single element of it on a bit of paper (possibly a very big bit of paper) without omitting anything. Is that a reasonable interpretation of your definition of a 'complete set'?

 

[matt grime]

Well, assuming you are using all those terms in the correct manner, the infinite binary tree has no leaves; there is not last level.

So perhas you ought to explain what all those things are. Given you didn't understand why a tree wasn't an uncountable totally disconnected set, it's very hard to even begin to say what you think any of these things are. That, and the fact you refuse to state what they are, of course.

By definition the root and leaf set of a tree is in the tree, when they exist: there is the infinite unrooted binary tree which has neither.

 

[Hurkyl]

Anyways, I would like to make my point about removing the leaves again:

One of these two diagrams is your "binary tree" with the lables removed.
The other of these two diagrams is your "binary tree" with the rightmost column chopped off, and the labels removed.
To draw each of them, I drew enough of the tree to represent the first 16 rows:

          /1
         1 
        / \0
       1   
       /\ /1
      /  0
     /    \0
 ... 1    
     \    /1
      \  1 
       \/ \0
       0  
        \ /1
         0
          \0
          
          /1
         1
        / \0
       1  
       /\ /1
      /  0 
     /    \0
 ... 0    
     \    /1
      \  1
       \/ \0
       0  
        \ /1
         0
          \0
 ...

          /1
         1 
        / \0
       1   
       /\ /1
      /  0
     /    \0
 ... 1    
     \    /1
      \  1 
       \/ \0
       0  
        \ /1
         0
          \0
          
          /1
         1
        / \0
       1  
       /\ /1
      /  0 
     /    \0
 ... 0    
     \    /1
      \  1
       \/ \0
       0  
        \ /1
         0
          \0
 ...

Can you tell which one is which? I can't.

 

[phoenixthoth]

seems like a fractal. at least it seems self-similar.

 

[Organic]

Can you tell which one is which? I can't.

If each notation represents more then one value then there is a difference between those trees (finite of infinte).

Again you counting the number of the notations and not the magnitude that each one of them representing.

If you don't remember then my answer to you was:

Your calculation is not right because:

1) aleph0 in my point of view stands for a general notation for any collection of infinitely many elements, and its value is flexible.

2) for example: By saying that a=(aleph0-aleph0) < b=aleph0 we mean that there are aleph0 elements in b that are not covered by a.

Also by a=(aleph0-2^aleph0) < b=aleph0 we mean that there are 2^aleph0 elements in b that are not covered by a.

There are no absolute magnitudes when we deal with collections of infinitely many elements, and no arithmetical operation (finite or infinite) can change their property of being infinitely many elements.

Shortly speaking, no arithmetical operation (finite or infinite) can change the “–“ or “+” sign in a collocation of infinitely many elements.

3) in my Binary tree the aleph0 width magnitude and the 2^aleph0 length magnitude, depends on each other, therefore their relative proportion (notated as width=aleph0 < length=2^aleph0) was not changed by your operation.


You still ignore the inner structure of infinitely many elements, because after your operation we have this list:

 {...,3,2,1}=N
     2 2 2
     ^ ^ ^
     | | |
     v v v
[b]{[/b]...,1,1,1[b]}[/b]<--> 1
 ...,1,1,1  
 ...,1,1,0 <--> 2 
 ...,1,1,0   
 ...,1,0,1 <--> 3 
 ...,1,0,1   
 ...,1,0,0 <--> 4 
 ...,1,0,0   
 ...,0,1,1 <--> 5 
 ...,0,1,1  
 ...,0,1,0 <--> 6
 ...,0,1,0  
 ...,0,0,1 <--> 7
 ...,0,0,1  
 ...,0,0,0 <--> 8
 ...,0,0,0  
 ...

So, as you see aleph0-1 < aleph0

By the way, the result of your oparation is ((aleph0)-1) < ((2^aleph0)-aleph0)and the reason that it is jusut -aleph0 and not -2^aleph0, can be clearly shown here:

 <---Arithmetic magnitude 

 {...,3,2,1,0} = Z*
     2 2 2 2  
     ^ ^ ^ ^   
     | | | |   
     v v v v  
{...,[b]1-1-1-1[/b]} <--> 1  Geometric magnitude(based on the 
 ...,1,1,1,[b]0[/b]  <--> 2          |          thin notations)          
 ...,1,1,[b]0[/b]/                   |
 ...,1,1/0,                   |
 ...,1,[b]0[/b], ,                   |
 ...,1/0, ,                   |
 ...,1|0, ,                   |
 ...,1|0, ,                   |
 ...,[b]0[/b]/ , ,                   |
 ...,0, , ,                   |
 ...,0, , ,                   |
 ...,0, , ,                   |
 ...,0, , ,                   |
 ...,0, , ,                   |
 ...,0, , ,                   |
 ...,0, , ,                   |
 ...                          V

After your operation we have:

( ((aleph0)-1) < ((2^aleph0)-aleph0) ) < ( aleph0 < 2^aleph0 )

Maybe this can help:

          /[b]1[/b]_1
         [b]1[/b]_2 
        / \[b]0[/b]_1
       [b]1[/b]_4   
       /\ /[b]1[/b]_1
      /  [b]0[/b]_2
     /    \[b]0[/b]_1
 ... [b]1[/b]_8    
     \    /[b]1[/b]_1
      \  [b]1[/b]_2 
       \/ \[b]0[/b]_1
       [b]0[/b]_4  
        \ /[b]1[/b]_1
         [b]0[/b]_2
          \[b]0[/b]_1
          
          /[b]1[/b]_1
         [b]1[/b]_2
        / \[b]0[/b]_1
       [b]1[/b]_4  
       /\ /[b]1[/b]_1
      /  [b]0[/b]_2 
     /    \[b]0[/b]_1
 ... [b]0[/b]_8    
     \    /[b]1[/b]_1
      \  [b]1[/b]_2
       \/ \[b]0[/b]_1
       [b]0[/b]_4  
        \ /[b]1[/b]_1
         [b]0[/b]_2
          \[b]0[/b]_1
 ...

          /[b]1[/b]_2
         [b]1[/b]_4 
        / \[b]0[/b]_2
       [b]1[/b]_8   
       /\ /[b]1[/b]_2
      /  [b]0[/b]_4
     /    \[b]0[/b]_2
 ... [b]1[/b]_16    
     \    /[b]1[/b]_2
      \  [b]1[/b]_4 
       \/ \[b]0[/b]_2
       [b]0[/b]_8  
        \ /[b]1[/b]_2
         [b]0[/b]_4
          \[b]0[/b]_2
          
          /[b]1[/b]_2
         [b]1[/b]_4
        / \[b]0[/b]_2
       [b]1[/b]_8  
       /\ /[b]1[/b]_2
      /  [b]0[/b]_4 
     /    \[b]0[/b]_2
 ... [b]0[/b]_16    
     \    /[b]1[/b]_2
      \  [b]1[/b]_4
       \/ \[b]0[/b]_2
       [b]0[/b]_8  
        \ /[b]1[/b]_2
         [b]0[/b]_4
          \[b]0[/b]_2
 ...

 

[moshek]

Dear Organic

What is the relation betwen your theoy to Frege fundaumanetal work fron 1878 that establish the foundation of mathematical Logic ?

Thank you
Moshe

 

[Organic]

Hi Moshek,

Please look here:

https://www.physicsforums.com/showthread.php?s=&threadid=16502&perpage=15&pagenumber=1

 

[moshek]

Thank you Organic !

There is allot of similarity to your concept of a a number to the way Prege develop first order logic but without the redundancy that you look on. So in some sense Prege work may be consider as a one example to your theory.

What about relativity theory of Ablert Einstein, since Prege was before Him ?

Thank you
Moshek

 

[Organic]

Moshek,

All I have now is Frege's basic information structures + uncertainty and redundancy as bulit-in properties of it.

Through my point of view these proprties have to be taken as basics properties of any modern theoretical research of any information system, like Math language for example.

 

[Organic]

After Albert Einstein there is no meaning to talk about space without time and vise versa.

Through my point of view there is no meaning to talk about quantity without structure and vise versa.



General conclusion:


The internal structure of any given quantity (finite or infinite) cannot be ignored.

 

[moshek]

Dear Organic :

The similarity and also the deferent
of your theory to Prege theory
is really amassing me !

Do you mean that Einstein develop new way
to look on the world but he still
was using Newton mathematics
and you suggest us and
alternativ mathematics ?

Moshek



www.icm2006.org

 

[Organic]

Hi Moshek,

Newton mathematics was a real break through that gave us the ability to deal with momentum in the real world.

Now it is about time to deal with complexity in the real world, but in my opinion it cannot be done if structural property of any mathematical product is not examined together with its quantitative property.

Shortly speaking any mathematical product is at least structural/quantitative product.

Ferge started to develop the connection between information's structure and logic, but its colleagues did not understand its attitude, ignored the information structure and developed only the private case of information structure of no_redundancy_no_uncertainty form.

 

[moshek]

Organic:

I think that Your are talking about mathematics without any
modeling or equations !

How all this is relate
to how Wittgenstein see mathematics?

Thank you
Moshek

 

[Hurkyl]

I forgot I had aborted my last reply.


If you'll allow me to use your analogy:

Einstein's great idea was that we should stop pretending we know the answer to "What IS the universe?" and focus on the question "What do we know ABOUT the universe?"


Modern mathematics uses the same idea; we don't care what things are, we only care about what we can do with them.


From a purely logical perspective, this is a phenomenal trick; if all of our theorems are based only on "What can we do with these things?", and are completely independant of "What are these things?", then all of our theorems are valid even if we have the wrong answer to "What are these things?"


This is why I think your approach is fundamentally flawed. You are focusing so hard on answering the question "What are these things?", but the question is irrelevant!

(And, for the record, I don't think your answer is even a valid one to this question)

 

[Organic]

This is why I think your approach is fundamentally flawed. You are focusing so hard on answering the question "What are these things?", but the question is irrelevant!

What is the connection between not ignoring the structural property
of the natural numbers and the question "What are these things?"

The inner structure of the natural numbers can be shown here:

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

 

[Organic]

Dear Hurkyl (this time please answer to this post),


My basic approach about the infinity concept is that redundancy and uncertainty are naturally involved, no more no less.

Therefore aleph0 is a notation that stands for general and flexible "cloud like" thing.

Cantor's thing is a frozen one, mine is not.

I think that my approach is much more interesting and fruitful than Cantor's approach.

Please let me put these two different approaches "on the table" and I would like to examine them together with you.

I am going to write my point of view on Cantor’s approach in a very simple way that (I hope) can be understood by you.

Please read it, and open my eyes to important things that I omit, don’t understand, distorting and so on.

So here it is:

1) Let set C be a complete non-empty binary tree where complete non-empty binary tree exists iff both root AND all its leafs are in C.

2) Let RT be the root , let LF be the all leafs,
therefore RT AND LF are in C --> [RT , LF].

( In my point of view RT XOR LF are in C --> ( (RT… , …LF) OR (LT… , LF] OR
[RT ,…LF) ) AND NOT [RT , LF] where “…” means unreachable. )

Now, from my point of view I see these basic problems when RT AND LF are in C:

1) If both RT AND LF are in C, then C must be a finite set.

2) If C is a non-finite set (through my point of view C is forced to be a non-finite set) then the base value 2 (which is the fundamental structural property of the non-empty binary-tree) cannot exist. Also RT value is unknown.

We must realize that if RT AND LF are in C AND C is a non-finite set, then the structural property of our information (in this case we are talking about the binary tree structure) collapsed into itself (we have no infinitely many elements anymore) and cannot be used as an input by Math language .

Therefore the expression 2^aleph0 cannot exist and we cannot construct
the transfinite universes.

Shortly speaking, what is called uncountable is not uncountable but simply does not exist in any form of input that can be used by Math language.

Through my point of view base 2 can exit
iff (C is finite) OR (RT XOR LF are in C)

Please show me how C is a non-finite set where
(RT AND LF are in C) AND (base 2 exists --> binary tree exists).

Also please tell me what is the value of RT when RT AND LF are in C AND C is a non-finite set.

 

[Hurkyl]

What is the connection between not ignoring the structural property
of the natural numbers and the question "What are these things?"

Exercise: try defining "structural property" without making any reference to what a natural number "is".

If both RT AND LF are in C, then C must be a finite set.

if RT AND LF are in C AND C is a non-finite set, then the structural property of our information ... collapsed into itself ... and cannot be used as an input by Math language

We math people call that a contradiction. And it CAN be used in "Math language" through the Law of contradiction:

"If P implies a contradiction, then P is false"

Or, more formally:

<br /> P \rightarrow (Q \wedge \neg Q) \vdash \neg P<br />

or equivalently

<br /> P \rightarrow Q, \neg Q \vdash \neg P<br />


Or, informally, it goes like this:

Let's make an assumption (call it P).
We derive a contradiction.
We conclude our assumption was wrong.

Or, if I may try and translate into Organic-speak:

If P causes our information to collapse, then P cannot be true.

Therefore the expression 2^aleph0 cannot exist and we cannot construct the transfinite universes.

And how does 2^aleph0 relate to anything above this statement in your post?

 

[Organic]

Exercise: try defining "structural property" without making any reference to what a natural number "is".

I did more then thet, it can be found here:
http://www.geocities.com/complementarytheory/count.pdf

You ignore the inner information structure of the natural numbers,
I don't.

Therefore your natural numbers are private case of my natural numbers
as can be clearly shown here:
http://www.geocities.com/complementarytheory/ETtable.pdf

Or, if I may try and translate into Organic-speak:

If P causes our information to collapse, then P cannot be true.

You cannot translate my ideas by your mathematical tools, because you ignore the information structure of the Binary tree as irrelevant to you, and looking only on its quantitative "shadow" that falling on the "real line".

And how does 2^aleph0 relate to anything above this statement in your post?

So you did not understand me then:

1) On what basis you translate me?

2) After you read all what I wrote can you answer to this?:

Please show me how C is a non-finite set where
(RT AND LF are in C) AND (base 2 exists --> binary tree exists).

Also please tell me what is the value of RT when RT AND LF are in C AND C is a non-finite set.

 

[Hurkyl]

You cannot translate my ideas by your mathematical tools

The reason for this is because you do not convey your ideas effectively, and you are self-contradictory.

(e.g. admitting that the set of real numbers satisfies the definition of "uncountable", yet in the same breath you assert that the set of real numbers is not "uncountable")

2) After you read all what I wrote can you answer to this?:

Please show me how C is a non-finite set where
(RT AND LF are in C) AND (base 2 exists --> binary tree exists).

Can you translate it into math-speak?

 

[moshek]

Hurkyl :

I there is in mathematics
a definition to "definition" ?

Thank you

Moshek

 

[Hurkyl]

I there is in mathematics
a definition to "definition" ?

I'm sure there are mathematical theories which have a class of objects called definitions, but in general the term "definition" is meta-mathematical, not mathematical, so to answer your question literally, the answer (in general) is no.

 

[moshek]

Dear Hurkl!

I was glad for your kind answer to me !

Since you are a matematition
but you can't defind "definition"
Way you ask it from Organic ?

Thank you
Moshek

 

[Hurkyl]

I'm not asking him to define definition, I'm asking him to provide one.


I'm probably somewhat more abstract than "mainstream" mathematics, but I would define "definition" metamathematically as simply one or more (precise) mathematical statements initially taken to be true.


The point, then, is that we can carry out logical deductions from the definitions to derive theorems.

 

[moshek]

No Hurkl !

The point is only a point !

Let me ask you my question in anothr way:

Do you believe that what happened to physice at 1905
chould Hapend also to mathematics in one day?

Thank you
Moshek

 

[Hurkyl]

It happened to mathematics first.

 

[moshek]

when ?

 

[Organic]

It happened to mathematics first.

If you speak about non-Euclidian geometry then you right but what about QM and Bohr's complementary attitude?

(e.g. admitting that the set of real numbers satisfies the definition of "uncountable", yet in the same breath you assert that the set of real numbers is not "uncountable")

No, N and R are enumerable by my definitions because my aleph0 is not your aleph0, no more no less.

Also I see that you don't understand my post about the difference between Cantor's aleph0 and my aleph0.

So, here it is again, but please this time stop on each part of it and please ask me about it, if you think that you don't understand it, thank you:

Dear Hurkyl (this time please answer to this post),


My basic approach about the infinity concept is that redundancy and uncertainty are naturally involved, no more no less.

Therefore aleph0 is a notation that stands for general and flexible "cloud like" thing.

Cantor's thing is a frozen one, mine is not.

I think that my approach is much more interesting and fruitful than Cantor's approach.

Please let me put these two different approaches "on the table" and I would like to examine them together with you.

I am going to write my point of view on Cantor’s approach in a very simple way that (I hope) can be understood by you.

Please read it, and open my eyes to important things that I omit, don’t understand, distorting and so on.

So here it is:

1) Let set C be a complete non-empty binary tree where complete non-empty binary tree exists iff both root AND all its leafs are in C.

2) Let RT be the root , let LF be the all leafs,
therefore RT AND LF are in C --> [RT , LF].

( In my point of view RT XOR LF are in C --> ( (RT… , …LF) OR (LT… , LF] OR
[RT ,…LF) ) AND NOT [RT , LF] where “…” means unreachable. )

Now, from my point of view I see these basic problems when RT AND LF are in C:

1) If both RT AND LF are in C, then C must be a finite set.

2) If C is a non-finite set (through my point of view C is forced to be a non-finite set) then the base value 2 (which is the fundamental structural property of the non-empty binary-tree) cannot exist. Also RT value is unknown.

We must realize that if RT AND LF are in C AND C is a non-finite set, then the structural property of our information (in this case we are talking about the binary tree structure) collapsed into itself (we have no infinitely many elements anymore) and cannot be used as an input by Math language .

Therefore the expression 2^aleph0 cannot exist and we cannot construct
the transfinite universes.

Shortly speaking, what is called uncountable is not uncountable but simply does not exist in any form of input that can be used by Math language.

Through my point of view base 2 can exit
iff (C is finite) OR (RT XOR LF are in C)

Please show me how C is a non-finite set where
(RT AND LF are in C) AND (base 2 exists --> binary tree exists).

Also please tell me what is the value of RT when RT AND LF are in C AND C is a non-finite set.

 

[Hurkyl]

I'm not talking about non-Euclidean geometry.

I'm talking about the idea of eliminating unnecessary assumptions.

Remember what SR was all about; it was proven that the speed of light is measured to be the same in all reference frames. Physicists of the time were trying to add new and mysterious things to physics to "save" their ideas of how the universe should work. Einstein said to heck with it and said that we might as well study what our measurements say.

My memory of timelines sucks, but I think mathematical formalism has several decades on Einstein. If not, it's in full swing now. *shrug* For instance, look at category theory; it cares nothing about the objects themselves, just the fact that they are objects, and that there are functions between objects.

 

[Organic]

Einstein said to heck with it and said that we might as well study what our measurements say.

What measurements?

Again, what about QM and Bohr's complementary attitude?

 

[Hurkyl]

What complementary attitude, and what about it?

 

[Organic]

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

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

 

[moshek]

Hurkyl :


I am sorry, but what happend
to physics at 1905 by Einstein
Did not happened yet
to mathematics!

Moshek

 

[matt grime]

Originally posted by moshek
Hurkyl :


I am sorry, but what happend
to physics at 1905 by Einstein
Did not happened yet
to mathematics!

Moshek

I can't quite decide *exactly* what happened in 1905 that's so important, but, Moshek, and please don't take this the wrong way, on what basis are you qualified to say what has happened to mathematics at any point?

 

[suyver]

Originally posted by matt grime
I can't quite decide *exactly* what happened in 1905 that's so important, but, Moshek, and please don't take this the wrong way, on what basis are you qualified to say what has happened to mathematics at any point?

1905 is Einsteins "magic year": it is the year in which he published special relativity, the resolution of the photoelectric effect (for which he would get the Nobel prize later on), and the theoretical description of Brownian motion (that almost proved beyond doubt the existence of atoms). These three theoretical discoveries gave a profound paradigm shift in physics.

I have no idea if such coincedences have also been seen in the field of mathematics (I am not an expert), but I guess they probably have (Euler, Gauss, other giants?)...

 

[moshek]

Dear Matt

Thank you for your nice question to me !

I will share with you today here very clear evidents that mathematics is standing today in front of changing paradigms like Einstein did in 1905 to Physics.

Dear Organic:

You may be interested in the following conference that stat today about the futher of Cardinals.

www.as.huji.ac.il/schools/math8/mathsprog.shtml[/URL]


Best
Moshek

 

[matt grime]

Arguably these are partly mathematical discoveries, at least I was taught about two of the three of them in my mathematics degree.

I suppose Russell would have to qualify, as would the category theory 'paradigm shift' (notice that, organic, shift, not change, shift) and then there was the bourbaki school, and the current shift away from that line of thinking. The mathematics of today is vastly different from that in Gauss's day - a lot of the proofs of that period don't stand up to scrutiny now. Then science was seen as a branch of philosophy, natural philosophy, and had according standards of proof. The shift in mathematics to today's view was gradual, and didn't have this alleged sudden epiphany, but it is arguable that saying physics fundamentally changed in 1905 is missing the years of research that went on up to that point that allowed these new discoveries to be accepted.

And we haven't even mentioned Cantor, Godel, Zermelo-Frankel, Nash, chaos theory, Mandelbrot, and various others to a lesser degree (Brylinski's loop space, Tsygan's simultaneous and independent discovery of Cyclic homology with Connes, Keller and Rickard's independent and simultaneous work on derived equivalences, Brauer's ground breaking work on representation theory...). What standards do you want to use to compare them?

 

[matt grime]

Originally posted by moshek Dear Organic:

You may be interested in the following conference that stat today about the futher of Cardinals.

www.as.huji.ac.il/schools/math8/mathsprog.shtml[/URL]


Best
Moshek [/QUOTE]


I doubt it, Organic has not shown the lsightest interest in learning anything about mathematics to tihs point, why should he start now.

 

[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

 

[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.