Monadic Mathematics

[Monad]

One of the efficient ways to change some framework is to ask new fundamental questions that maybe lead us to new frontiers.

For example, the fundamental question of the language of Mathematics framework is: ”How many?”.

Let us try another question, for example: “What do we have?”

A closer look of these questions shows that ”How many?” questions are mostly about the Quantity concept, where “What do we have?” questions are mostly about the Structure concept.

Let us check if “What do we have?” questions can be a fruitful ground for mathematical development.

First we have to define the minimal concepts that can be used under the structure concept.

In other words, these concepts when used, determine our framework’s domain, for example:

Let use say that the two main concepts that are related to the structure concept are Length and Direction.

It means that by using these concepts, we can define the building-blocks of our mathematical framework.

We also know that by using these building-blocks we suppose to get some input that can be used by us to develop our framework.

So, the next question is “What are the limits that beyond them no input can be found?”

The lowest “no input” state is Emptiness, where Length and Direction do not exist.

The highest “no input” state is Fullness, where Length and Direction are beyond measurement.

So the useful elements have measurable Length and/or Direction, that enable us to use them as some input.

Let us represent these ideas in a table:

          Monadic Mathematics build-blocks        
          --------------------------------

V = Available

X = Not Available
        
          +-----------------------------+
          |       Measurement of        |
          |                             |
          |     Length   |   Direction  |
          |--------------|--------------|
          |              |              |
       {} |              |              | Emptiness
          |              |              |
          |              |              |
          |--------------|--------------|
          |              |              |
      {.} |       V      |              | Point
          |              |              |
          |              |              |
          |--------------|--------------|
          |              |              |
    {._.} |       V      |       V      | Segment
          |              |              |
          |              |              |
          |--------------|--------------|
          |              |              |
          |              |       V      | Tendency
          |              |              |
          |              |              |
          |--------------|--------------|
          |              |              |
     {__} |       X      |       X      | Fullness
          |              |              |
          |              |              |
          +-----------------------------+

General:

{' and '}' are the notations of a framework which we call a set.

Between these notations we can put our examined concepts and then try to find out what we can do with each one of these concepts, and also what interactions can be found between concepts and themselves and/or concepts with other concepts.

The concept of Nothingness or Emptiness is notated in this framework by {}.

Any examined concept in my framework is examined by its structural properties and also by its Quantitative properties.

The most basic Structural property in my framework is based on the Length concept.

From this point of view, a Point {.} has exactly 0 Length.

The Length concept has no meaning in my framework when it is related to the Emptiness concept (which is notated as {}), therefore 0 cannot be connected directly to the Emptiness concept.

Another option to define 0 is to ask: How many things there are in {}?

By 'how many?' question we actually define the Cardinal concept, which is notated by using '|' and '|' .

In this case, the cardinal of {}, which is notated as |{}|, is equal to 0.

In Standard Mathematics framework (when Ordinality is omitted) the one and only one option to connect between the Number concept and the Set concept, is by using only the Quantity concept,
and in this case |{}| = 0.

But as you see, in Monadic Mathematics framework, there are two kinds of cardinals, where one of them is the standard Quantitative Cardinal, but the second type of cardinal is what I call the Urelement Cardinal, which is based on the Length concept (which is the structural property of the Number concept).

In Monadic Mathematics, the structural property of a number is more basic then its quantitative property.
According to what I wrote above, when a Point eliminates itself, then the result is Emptiness.

In other words Point - Point = Emptiness, or in other representation:
0 - 0 = {}.

Standard Mathematics takes '0' notation as something which is first of all related to the Quantity concept.

In this framework 0 - 0 = 0

Monadic Mathematics takes '0' notation as something which is first of all related to the Structure concept.

In this framework 0 - 0 = {}




A non technical explanation of Monadic Mathematics' '+' and '-' operations:

The most basic question of Standard Mathematics is “How many?”.

The most basic question of Monadic Mathematics is “What do we have?”

The basis of a “How many?” question is the Quantity concept.

The basis of a “What do we have?” question is the Structure concept.

It means that the Number concept in Monadic Mathematics, is first of all based on the Structure concept.

Question:
What are the minimal, and distinguished, structural forms in Monadic Mathematics?

Answer:
Emptiness (notated as {}), Point (notated as {.}), Segment (notated as {._.}), Fullness (notated as {__}).

{} and {__} are the weak ({}) and the strong ({__}) limits of Monadic Mathematics framework.
It means that they cannot be used as inputs in Monadic Mathematics.

So, Monadic Mathematics operations are based on {.} and {._.}.

The two basic structural properties, which related to the Number concept, are Length and/or Direction.

{.} has 0 length and no directions.

{._.} has 0_x length and at least two opposite directions, which are
0_x or x_0.

The most basic arithmetical operations between these elements are ‘+’ and ‘-‘.


Important:

If '-' sign is used not as a binary operation, for example: -x_0, then it is a negation symbol and not a binary Elimination operation.

Therefore -x_0 = 0_x or -0_x = x_0.

Also {} - x_0, which is a vacuous binary operation (because {} or {__} cannot be used as inputs) is equal to -x_0 = 0_x, etc.

‘+’ is understood as Concatenation, for example:

1_0 + 1_0 = 2__0 (._. + ._. = .__.)

0_1 + 1_0 = 1_0_1 (._. + ._. = ._._.)

0 + 0 = 0 ( . + . = . because Concatenation of 0 Length is 0 Length)

0__2 + 0_1 = 0___3 (.__. + ._. = .___.)

2__0 + 1_0 = 3___0 (.__. + ._. = .___.)

0_1 + 2__0 = 2__0_1 (._. + .__. = .__._.)

etc …


‘-’ is understood as Elimination, for example:

1_0 - 1_0 = {} (._. - ._. = {} because both Length and Direction are identical)

0_1 - 1_0 = 0 (._. - ._. = . because only Length is identical)

0 - 0 = {} ( . - . = {} because Length is identical and there is no Direction)

0__2 - 0_1 = 0_1 (.__. - ._. = ._.)

2__0 - 1_0 = 1_0 (.__. - ._. = ._.)

0_1 - 2__0 = 1_0 (._. - .__. = ._.)

etc …


More detailed information can be found in:

http://www.geocities.com/complementa...rst-axioms.pdf


According to Monadic Mathematics the Number concept is the fundamental building-block of a non-destructive interaction between opposites, which is based on Included-middle reasoning (which is the logic that can be found between at least two opposites that define their middle domain).

For more details please read:

http://www.geocities.com/complementa...loisDialog.pdf

[Keeblergiant]

I would recommend nobody pay attention to this guy. All he does is invade message boards, post "his" mathematics (it's really just previously known stuff with a different notation), and argue. It's more philosophy than mathematics anyways.

[Red Shift]

wha?........ ¬_¬

hehe never been any good at maths, i don't mean to start now ;)

[Monad]

Originally Posted by Keeblergiant

it's really just previously known stuff with a different notation

Dear Keeblergiant please give me an address where I can find this "previously known stuff".

Thank you.

[Keeblergiant]

Address? I didn't know that basic set theory and geometry had addresses? Did you skip 9th grade math or something?

[Keeblergiant]

But, if you believe that your ideas are original, would you mind giving me an example of something original about your ideas?

[Monad]

Dear Keeblergiant,

In MM (Monadic Mathematics) the Natural number > 1 is based on complementary relations between Set_AND_Multiset, where its internal structure is ordered by complementary relations between multiplication and addition operations.

For more details please look at:

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

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

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

[Keeblergiant]

Set theory (not basic set theory, but set theory nonetheless).

[Monad]

Excuse me dear Keeblergiant,

Where are the details of your reply?

[Keeblergiant]

It's pretty much just like you said...the natural numbers can be derived by the cardinals of sets and the multiplicity of the cardinals of those sets in a multiset, and you can determine order using binary operations.

[Monad]

In my system I have two kinds of cardinals:

1)Quantitative cardinal.

2)Urelement cardinal.

Also I have to kinds of ordinals:

1) External ordinal.

2) Internal ordinal

Please look at http://www.geocities.com/complementa...rst-axioms.pdf to understand this.

Thank you.

[Keeblergiant]

Your quantitative cardinal is the same as the "real" cardinal. And since your {a___b} is analogous to {[a,b]}, with the rule that [a,b] be considered one object, your urelements would be max[a,b] and min[a,b]. I didn't see anything in the link you provided about your ordinals, but if you'll send me a link to them, I'll be sure to tell you how you are just putting a different shirt on modern mathematics and calling it your creation

[Monad]

Originally Posted by kiblergiant

Your quantitative cardinal is the same as the "real" cardinal. And since your {a___b} is analogous to {[a,b]}, with the rule that [a,b] be considered one object, your urelements would be max[a,b] and min[a,b]. I didn't see anything in the link you provided about your ordinals, but if you'll send me a link to them, I'll be sure to tell you how you are just putting a different shirt on modern mathematics and calling it your creation

No, 0_x is a singleton and a non-composed element where [a,b] is not a singleton and a composed element.

Actually, [a,b] is the opposite structure of 0_x, and is considered as not less than a pair in Standard Mathematics.



Some examples of my work:

------------------------------------------------------------------------------
If my system is analogous to modern mathematics, then please show us in what mathematical branch we get this?:

0_x - 0_x = {} (Empty set)

0_x/0 = {__} (Full set)

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

------------------------------------------------------------------------------
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/complementa...rst-axioms.pdf

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

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

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

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

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

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

-------------------------------------------------------------------------------
A new approach about Russell's first paradox, can be found here:

http://www.geocities.com/complementa...y/Russell1.pdf

-------------------------------------------------------------------------------
A new approach about Cantor's diagonal methods can be found here:

http://www.geocities.com/complementa...agonalView.pdf

-------------------------------------------------------------------------------
A new approach about Collatz' problem can be found here:

http://www.geocities.com/complementa...y/3n1proof.pdf

-------------------------------------------------------------------------------
A new approach about the Real numbers can be found here:

http://www.geocities.com/complementa...Naive-Math.pdf

-------------------------------------------------------------------------------
A new approach about the Infinity concept can be found here:

http://www.geocities.com/complementa...annsLimits.pdf

-------------------------------------------------------------------------------
A new approach about the Function concept can be found here:

http://www.iidb.org/vbb/showthread.php?t=102717

-------------------------------------------------------------------------------
A new approach about the Logic concept can be found here:

http://www.geocities.com/complementa.../CompLogic.pdf

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

[Keeblergiant]

0_x - 0_x = {} (Empty set)

0_x/0 = {__} (Full set)

a-a=0, and a/h -> infinity as h -> 0 (from the positive direction). Therefore, your "ray" (can't remember what you call it again), would encompass every positive real number, and be a full set. That is 1st grade math and beginning calculus, respectively. And notice that I said that the object [a,b] be considered one object in my explanation, so it would still be considered a singleton. And that is still irrelevant, as your urelement cardinal is still analogous to the min and max functions. I'm not reading the rest of that crap, I have stuff to do besides spend an hour reading a bunch of altered-notation mathematics from some [edit] in Isreal.

[Monad]

Originally Posted by Keeblergiant

a-a=0, and a/h -> infinity as h -> 0 (from the positive direction). Therefore, your "ray" (can't remember what you call it again), would encompass every positive real number, and be a full set. That is 1st grade math and beginning calculus, respectively. And notice that I said that the object [a,b] be considered one object in my explanation, so it would still be considered a singleton. And that is still irrelevant, as your urelement cardinal is still analogous to the min and max functions. I'm not reading the rest of that crap, I have stuff to do besides spend an hour reading a bunch of altered-notation mathematics from some loon in Isreal.

So, now we discover that you are not just one that cannot uderstend what is a Urelement, cannot distinguish between 0 and {} or composed element like [a,b] and non-composed element like 0_x, or positive R members and {__} (The full set), but you also have a rude personality.

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

Some analogy that can help to uderstand Included-middle reasoning:

Let us say that the property of being a glue depends on some mixing between
material A and material B.

A and B are not defined (exist) by each other, but when they are mixed together in a particular way, we get the property of a glue.

The most comprehensive principle of this point of view is based on the middle domain that can be found between at least two opposites, when the property of the middle domain is a new thing that cannot be found in each separate opposite.

And this is exactly the main principle of Included-middle logical reasoning, which define most of the time new products, which are the outcome of at least two opposites.

In other words, any non-trivial abstract/non-abstract product is first of all based on constructive associations between opposites as their first-order products, and then the rest of the middle domain elements, are some n>1 order products.

[Keeblergiant]

So, now we discover that you are not just one that cannot uderstend what is a Urelement, cannot distinguish between 0 and {} or composed element like [a,b] and non-composed element like 0_x, or positive R members and {__} (The full set), but you also have a rude personality.

I know perfectly well what a urelement is, I can distinguish between 0 and {} and [a,b] and 0_x, and positive R members and {___}. The point is is that they are all analogous, and they provide the same answers as your mathematics, but they use a more common and useful notation, so your mathematics provide nothing new or advantageous. And I am quite rude when morons start making shit up and showing it off as though it's the next best thing since sliced bread. Now, my recommendation for you is to get an education, discuss your "ideas" with real mathematicians, and see how they hold up. Obviously they don't hold up too well as of yet, because you've been rejected from mathematics journals before.

[Keeblergiant]

Oh, and I've also found many holes in your reasoning...I'll post them next time I get the chance...1:40 is too late for me to do any serious writing.

[Monad]

Originally Posted by Keeblergiant

I know perfectly well what a urelement is, I can distinguish between 0 and {} and [a,b] and 0_x, and positive R members and {___}. The point is is that they are all analogous,

One of the nice things in any smart system is the basic property of feedback.

From what you wrote above it is easily understood that you have some severe problems to use this property, for example:

You do not understand the total logical failure of what you wrote, because (for example) 0 is totally different from {} and because of this difference we can clearly distinguish between them, therefore they cannot be analogous.

By your logical reasoning distinguish=analogous, and it is easily understood that your motives to post to me are based only on bad emotions.

Originally Posted by Keeblergiant

I'm not reading the rest of that crap, I have stuff to do besides spend an hour reading a bunch of altered-notation mathematics from some loon in Isreal.

First, the name of my beautiful country is Israel, and not Isreal (but indeed it IsReal great place to live in).

Second, if you write

Originally Posted by Keeblergiant

Oh, and I've also found many holes in your reasoning...I'll post them next time I get the chance...1:40 is too late for me to do any serious writing.

then it is again clearly understood that you don't distinguish that you contradict yourself again, because after all, you took the time to read "the rest of my crap", which is again clearly motivated by emotions and not by logical reasoning.

Any way, even in this poor condition of yours, I would like to see the logical holes that you have found in my work, because there is nothing like some good criticism as a canalization for some framework development (which in your case I am not sure that you have the right properties to do that).

[Keeblergiant]

You do not understand the total logical failure of what you wrote, because (for example) 0 is totally different from {} and because of this difference we can clearly distinguish between them, therefore they cannot be analogous.

NO! In this situation, they are analogous. Think about this: We have a "point" {a}. Now, if we think of such in terms of just numbers, we have a-a=0. However, if we think of such in terms of set theory, we have {a}\{a}={}, as "-" and "\" are analogous operators.

And as for my motives to post being driven by emotion instead of logic, that is irrelevant and it has nothing to do with the fact that I'm right.

Now for the problems with your mathematics (note, I'm only using two pages out of the whole thing right now, if you wish for more, let me know):

All of the quoted sections are from your paper.

A singleton set s that can be defined by tautology* ('=') and ('<' or '>'), where s has no internal parts

The '=', '<', and '>' "signs" are used to define the internal parts of an interval, so by definition, it is a contradiction.

(In standard Math we had to write: A point: If a content of a set is a singleton and a urelement and has no directions, then it is a point; A segment: If a content of a set is a singleton and a urelement and also has directions, then it is a segment.)

Nothing can be defined a singleton and a urelement, because a singleton is described as a set containing one element, and part of the definition of a urelement is that it contains no elements.

My critique of your axiom of independency is that it is not an axiom unless you can provide a logical reason as to why it is an axiom (I refer you to Euclid's Elements). I also have a problem with the content of the axiom, as both logic and your notation both imply that an interval is defined in terms of two points. In fact, if you think about it, you CANNOT determine an interval without points if we use your definition of an interval.

My critique of your axiom of minimal structure is that it is not an axiom, as there is no logical reason for it to be so. Once again, I refer you to Euclid's Elements.

There is no logical basis for your axiom of duality either.

And, your method did not define the set R, although this is not a problem because you can apply the same method of Dedekind cuts to determine the real number set.

That's all I'm doing right now, ask for more if you want. Until then, shut up.

[Monad]

Wow, you really have big problems to use simple mathematical concepts, for example:

1) the closed interval [a,b] is based on the pair {a,b} where |{a,b}|=2

2) |{0_x}|=1

So as you can see: 0_x is not [a,b].

You can clime until the end of time that [a,b] can be considered as a one element, end you can prove it only if you prove that 1=2.

[Keeblergiant]

You are obviously not understanding my post. I said that if we treat [a,b] as one object. Therefore, by definition |{[a,b]}|=1. It's that simple. And, education has nothing to do with human behavior. I'm a fine person when ignorant morons aren't pissing me off. YOU'RE NOT A GENIUS, AND YOUR MATHEMATICS ARE NEITHER NEW NOR INNOVATIVE. So, once again, the education recommendation holds.

[Keeblergiant]

You can clime until the end of time that [a,b] can be considered as a one element, end you can prove it only if you prove that 1=2.

That's funny, as your mathematics seems to depend on intervals being treated as one object.

[Monad]

Originally Posted by Keeblergiant

my recommendation for you is to get an education...

According to your last posts you are not in any position the recommend anything, not about Mathematics and not about human behavior.

I'll continue soon...

Do you know? in my system 0 - 0 = {} and 0 + 0 = 0.

Now, please explain it by Stantard Math.

Originally Posted by Keeblergiant

The '=', '<', and '>' "signs" are used to define the internal parts of an interval, so by definition, it is a contradiction.

Well this is another example of your inability to understand what you read.

In my system the concept of an interval is exactly the opposite of the standard meaning.

It is based on indivisible (non-composed) single segment, and this segment has 3 basic properties:

a) It is an indivisible (non-composed) single element (Singleton_AND_Urelement).

b) It has a finite length {._.}

c) It has a direction, '<' = 'left-right' = '0_x' or '>' = 'right-left' = 'x_0'

In my system no form of {._.} is composed by shorter forms of {._.} or inifinitely many {.} forms.

Please read at least pages 13-14 of http://www.geocities.com/complementa...rst-axioms.pdf to understand this new point of view.

Originally Posted by Keeblergiant

Nothing can be defined a singleton and a urelement...

Wrong again, this is exactly an indivisible non-composed and one solid segment {._.}.

Originally Posted by Keeblergiant

I refer you to Euclid's Elements

It is irrelevant for my framework, which is based on a Non-Euclidian school of thought, which you clearly show that you do not grasp it (yet).

At this stage you are not in any position to understand my framework, because you try to understand it from an Euclidian point of view.

Originally Posted by Keeblergiant

Therefore, by definition |{[a,b]}|=1

Even this clime of you is irrelevant because [a,b] is a composed element, where ._. is a singleton AND a Urelement.

Originally Posted by Keeblergiant

That's funny, as your mathematics seems to depend on intervals being treated as one object.

In my framework you can find both the standard system and the new system.

Please look at pages 18-20 in http://www.geocities.com/complementa...rst-axioms.pdf in order to see it by yourself.

[Keeblergiant]

Do you know? in my system 0 - 0 = {} and 0 + 0 = 0.

Send me a link to the paper that that is shown, as your notation is so mangled and distorted I do not know what each claims.

Wrong again, this is exactly an indivisible non-composed and one solid segment {._.}.

Wrong. You should check mathworld or wikipedia.

Well this is another example of your inability to understand what you read.

In my system the concept of an interval is exactly the opposite of the standard meaning.

It is based on indivisible (non-composed) single segment, and this segment has 3 basic properties:

a) It is an indivisible (non-composed) single element (Singleton_AND_Urelement).

b) It has a finite length {._.}

c) It has a direction, '<' = 'left-right' = '0_x' or '>' = 'right-left' = 'x_0'

In my system no form of {._.} is composed by shorter forms of {._.} or inifinitely many {.} forms.

Your paper said nothing about defining the "signs" as you just have.

It is irrelevant for my framework, which is based on a Non-Euclidian school of thought, which you clearly show that you do not grasp it (yet).

THERE IS NOTHING NON-EUCLIDEAN ABOUT YOUR PAPER! And besides that, I wasn't referring to the elements for the geometry aspect, but for how axioms and postulates should be based. But, back to your framework being Non-Euclidean...how is that so? Do you even understand the concept of non-Euclidean geometry? The most fundamental aspect of non-Euclidean geometry is the metric tensor, yet you do not even mention it. Quit making up false definitions for words such as non-Euclidean, as they already have meaning and you will confuse those who do not know that you have no clue what you are talking about. And it is relevant to your framework, because your framework is based on falsely claimed axioms.

Even this clime of you is irrelevant because [a,b] is a composed element, where ._. is a singleton AND a Urelement.

In my example, I was treating the interval as a singleton (to use your terms), just to show you that your cardinal was nothing new. AND A SINGLETON CANNOT BE A URELEMENT. A URELEMENT BY DEFINITION CONTAINS TO NO ELEMENTS, WHERE AS A SINGELTON BY DEFINITION CONTAINS ONE ELEMENT!

[Monad]

I wasn't referring to the elements for the geometry aspect

I am not speaking about Non-Euclidian Geometry, but about Non-Euclidian Mathematics.

Non-Euclidian Mathematics' elements cannot be found in any Encyclopedia (yet) because it is a new and fundamental point of view of the Language of Mathematics and its logical reasoning, which is based on Liebniz' Monads idea (which are understood as {}, {.}, {._.} , {__} four building-blocks of my Monadic Mathematics) and on include-Middle logical reasoning ( http://www.geocities.com/complementa.../CompLogic.pdf )

Wrong. You should check mathworld or wikipedia.

This is exactly what I did, and it fits perfectly to the Liebliz' Monad's concept that I use in my framework.

A URELEMENT BY DEFINITION CONTAINS NO ELEMENTS

Yes, I agree with you, but you missed the fine meaning of this definition.

The meaning is: The Urelement contains no elements, which means, it is a non-composed indivisible one-solid element.

In short, you missed the fine and deep meaning of the letter 's' in the end of the word 'elements'.

"An Urelement contains no elements, belongs to some set, and is not identical with the empty set" ( http://mathworld.wolfram.com/Urelement.html )

By this definition we get a one solid (non-composed) element.

In short, each one of {.}, {._.} or {__}, is an Independent type of a Urelement.