Structure and Quantity

November 4, 2004

Any number can be distinguished form another number by at least two basic properties, which are:

a) Structure

b) Quantity

Let me show you a nice thing, for example, let us take number 26.

Now we can show that this quantity has more than one internal structure, which is based on the base value that we are using:

Number 26 represented by base 10:

         ^0- 0123456789
             ||||||||||
             |_||||||||
             |__|||||||
             |___||||||
Base 10 =     |____|||||
             |_____||||
             |______|||
             |_______||
             |________|
         ^1- |
             |                1      0
                        ( [color=Magenta][b]2[/b][/color]*10 + [color=Magenta][b]6[/b][/color]*10 )
^0- 01234567890123456789012345[color=Magenta][b]6[/b][/color]
   |||||||||||||||||||||||||||
   |_|||||||||_|||||||||_|||||
   |__||||||||__||||||||__||||
   |___|||||||___|||||||___|||
   |____||||||____||||||____||
   |_____|||||_____|||||_____|
   |______||||______||||__...
   |_______|||_______|||__...
   |________||________||__...
^1- |    0    |    1    |  [color=Magenta][b]2[/b][/color]
   |_________|         |   
   |___________________|   
   |_ ...
   |



Number 26 represented by base 3:

         ^0- 012
Base 3 =      |||
             |_|
         ^1- |         3     2     1     0
                  ( [color=Magenta][b]0[/b][/color]*3 + [color=Magenta][b]2[/b][/color]*3 + [color=Magenta][b]2[/b][/color]*3 + [color=Magenta][b]2[/b][/color]*3 )
^0- 01201201201201201201201201[color=Magenta][b]2[/b][/color]
   |||||||||||||||||||||||||||
   |_||_||_||_||_||_||_||_||_|
^1- |0 |1 |2 |0 |1 |2 |0 |1 |[color=Magenta][b]2[/b][/color] 
   |__|  |  |__|  |  |__|  |  
   |_____|  |_____|  |_____|  
^2- |  0     |  1     |  [color=Magenta][b]2[/b][/color]     
   |________|        |        
   |_________________|        
^3- |        [color=Magenta][b]0[/b][/color]                 
   |_ ...
   |

So, as you can see, even the same quantity can be distinguished by at least two different internal fractal structures.

From this example we can understand that the concept of a Number is some information form, which is based on not less than Structural_AND_Quantitative properties.

Sayonara · November 4, 2004

Is that significant?

November 4, 2004

Is that significant?

If we look at the Language of Mathematics as an information system, then no kind of potential information can be totally ignored.

It means, that if we want to use the Number concept as a model that holds also the structure of the information, and not just the quantity of it, then the internal structure of the Number concept is important and must not be ignored.

For example, proofs by induction that are based only on the quantitative aspect of the Number concept, cannot work if each quantity has also several internal information structures.

In this case we have to choose (or examine) the internal structure of each inductive step, and then we can understand that a lot of, so called, proofs by induction, are nothing but some trivial results that actually ignore important aspects the complexity that can be found in the examined problem.

Ophiolite · November 4, 2004

Can you give a simple example? One that several of us, not all necessarily gifted in mathematics, might understand. As my signature may imply there is more data and information than we can reasonably process. There has to be a filtering mechanism. Sayonara's relevance question is part of such a mechanism. I appreciate that you have replied to that, but without a concrete example I am afraid the explanation went over my head.

matt grime · November 4, 2004

"proofs by induction that are based only on the quantitative aspect of the Number concept"

No they aren't. Proofs by induction are based upon the well-ordering principal, and do not *necessarily* make any reference to quantity, though in the natural numbers the well ordering statement is equivalent to the existence of a least element in any non-empty set, but it is not a necessary condition, merely a sufficient condition.

That doesn't mean one cannot do induction without the natural numbers, nor does it mean that if we do not "examine" the "structure" of the natural numbers, whatever that may be, that proofs by induction fail to be true. Your "structure" concept has no bearing on proof by induction since all the properties of induction are only dependent on well-ordering. If you are about to redefine the concept of natural numbers so as they aren't well ordered, then you cannot talk about induction: it doesn't make it false since you are changing the premise on which it is founded and not using a well ordered set.

November 4, 2004

Hi Matt and Ophiolite,

Matt, First thank you for your beautiful and clear post.

If we take only the Natural numbers, then what comes first Quantity or Order?

As much as I understand it, a well-ordering principal of the Natural numbers is based on the Quantity concept, isn't it?

In short, we first recognize some N member by its quantitative property, and only then we can decide in what order it will appear in some collection.

In other words, each N member in some collection is first of all distinguished by its quantitative property.

It means that the well-ordering principal in Standard Math cannot be expressed without the Quantity concept, in the case of the Natural numbers.

The same state also holds in my system (Monadic Mathematics) but since in my number system any Organic Natural Number also has an internal structure, then to any given quantity there are several distinguished internal structures, that can be ordered by their symmetrical degrees, which are based on complementary relations between Multiplication and Addition operations.

Organic Natural Numbers can be shown here:

http://www.geocities.com/complementarytheory/ONN1.pdf

http://www.geocities.com/complementarytheory/ONN2.pdf

http://www.geocities.com/complementarytheory/ONN3.pdf

and their internal structures are an extension of the well-ordering principle.

Since I show that N members are only some single structural case of Organic Natural Numbers, then from this deeper point of view of the Natural number concept, a lot of proofs by induction of Standard Math cannot hold, when the internal structure of each given quantity is not ignored.

Please take a look at page 20 of http://www.geocities.com/complementarytheory/ONN3.pdf where we can clearly see that Fibonacci sequence cannot be fully defined by using the standard N members.

And the reason that Fibonacci sequence cannot be fully defined is that we have to choose one of several possible internal structures, in order to define the particular Fibonacci sequence.

In other words, from this point of view there are infinitely many Fibonacci sequences that simply ignored when each N member has one and only one internal structure, which is based only on 0_Redundancy_AND_0_Uncertainty information form.

matt grime · November 5, 2004

Doron, you are once more moving the goal posts. The proofs hold still because they are not written about *your* objects. If you change the meaning of all the terms in some proposition, then of course it will almost certainly fail to be true, but you don't get to do that and then tell others they are wrong. All that remains is in *your* system (one of so far no demonstrable value, in fact the exact opposite since so many things are false) the statement is not true.

Truth depends only on the context, there is no absolute truth in mathematics.

As a simple example of why your observation might not be true anyway: hte natural numbers possess an arithmetic, and we may define primes in them. I do not at any point use these properties in an inductive proof : that doesn't mean the induction is invalid in the system in which I am operating.

November 5, 2004

but you don't get to do that and then tell others they are wrong

I am not talking about 'right' or 'wrong'' date=' this is [b']your[/b] interpretation of post #6. (maybe, just maybe this is your way to interpret of my posts, because you use 0_XOR_1 logical reasoning to analyze my posts, where my logical reasoning is based on the Included-Middle point of view, where two opposites simultaneously preventing/defining their middle domain).

What I show is that there is a deeper point of view of the same elements, where the "old" proofs become trivial (trivial not in the mathematical meaning) because they ignore the internal complexity of the examined elements, and in this case I am talking about N elements, which I clearly show that they are some single case of my number system.

In short dear Matt, when I say that some proofs of the Standard point of view do not hold, I do not think in terms of 'right'_XOR_'wrong', but in terms of the triviality and non-interesting results of these proofs, when they are understood from a non-trivial point of view.

...natural numbers possess an arithmetic...

And I clearly an simply show the advantage of my system' date=' when it is compared to the Standard '+','-' arithmetic.

This comparison can be found in pages 22-29 of http://www.geocities.com/complementarytheory/My-first-axioms.pdf

Truth depends only on the context, there is no absolute truth in mathematics.

I am totally agree with this point of view, and because of this point of view fundamental elements like N members can be changed, when they are observed from a deeper and richer point of view.

matt grime · November 5, 2004

You see that's where everyone but you disagrees. You don't get to change N. You can create whatever you care for, but it (probably) won't replace the naturals as we know them. We have a simple set of rules for the naturals, and there is no need to change them. You may create something new and different, but stop trying to sell them as "the natural numbers", because they aren't. Sometimes definitions do change over time, for many different reasons: some used to consider 1 a prime, now we do not. I doubt that you are going to change the definition of the naturals.

November 5, 2004

We have a simple set of rules for the naturals' date=' and there is no need to change them.

[/quote']

Please explain us why (if N is clearly some single case of my ONN)?

premjan · November 12, 2004

I kind of like your framework of monadic mathematics in that it starts with a larger set of irreducible geometric and algebraic properties (e.g. fullness) so that it is probably a little easier to justify each step why. However, your notion of attaching the "fractal" representation of a number to the number itself seems little more than a half-step towards a computer algorithm. Why not just take the full step and talk just in terms of Numerical Algorithms? Maybe you have some good reason for preferring what seems to me to be a kind of halfway house. Who knows you may even be onto something, and future pedagogy may be based on some rather composite notions such as yours rather than the existing system of proofs which is a little leaner in my estimation.

Good luck getting someone to adopt your system, in say, a school, since that is where you might have to start to get off the ground.

November 12, 2004

Dear premjan,

First we need a Hardware/Software environment, which its architecture is based on my framework, in order to use my framework.

In short, pencil and paper are not enough any more in order to deal with the full capacity of Monadic Mathematics and its logical reasoning.

Nobody can draw the Mandelbrot Set without using a computer, and in the same manner, my system needs a new environment in order to fully work.

Its basics are simple but its products are very complex, and most of them belong to the future.

premjan · November 12, 2004

...computer architecture to support monadic mathematics...

if you're into programming, you could (or you could hire) someone to implement this sort of thing. I don't think computer numeric implementations depend on things like fullness at all, so it would really only show up at the computer algebra package level (you might consider starting off your own software package, call it neoCantor or something like that).

Unfortunately I do not appear to be the right person to help you in this regard. But good luck.

November 12, 2004

But good luck

Thank you dear premjan.

I am talking about changing the Binary Turing machine principle, to an ONN machine.

premjan · November 12, 2004

Thank you dear premjan.
I am talking about changing the Binary Turing machine principle' date=' to an ONN machine.[/quote']

OK, now I begin to get it. There's apparently a whole branch of logic called Monadic logic which is not the party currently in power. I bet it is a bit smarter and more efficient than the standard dyadic stuff. In my experience dyadic stuff (e.g. Chrisitianity) always precedes monadic stuff (e.g. Islam). There I go spouting useless philosophical abstractions now.

Are you really a logician? I don't care if you are self-taught. Hardly matters.

November 13, 2004

Some examples of my work:

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A proof that cannot be accomplished by using 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

To understand this proof, please read at least page 13 of http://www.geocities.com/complementarytheory/ONN2.pdf

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A test that shows the advantage of - and + operations in an included-middle logical reasoning framework, can be found in pages 22-29 of http://www.geocities.com/complementarytheory/My-first-axioms.pdf

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Complementary relations between Multiplication and Addition binary operations can be found in pages 7-8 of http://www.geocities.com/complementarytheory/ONN1.pdf

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A fundamental new approach about the Natural numbers can be found in:

http://www.geocities.com/complementarytheory/ONN1.pdf

http://www.geocities.com/complementarytheory/ONN2.pdf

http://www.geocities.com/complementarytheory/ONN3.pdf

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A new approach about 0.9999... = 1 can be found here:

http://www.geocities.com/complementarytheory/9999.pdf

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A new approach about the Limit concept can be found here:

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

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