Doron Shadmi
http://www.tahal.com
Israel
Registered: Aug 2004
Posts: 9 
Monadic Mathematics
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 buildingblocks of our mathematical framework.
We also know that by using these buildingblocks 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:
code:
Monadic Mathematics buildblocks

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 implies Emptiness, or in other representation:
0  0 implies {}.
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 implies {}
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 implies {} (._.  ._. implies {} because both Length and Direction are identical)
0_1  1_0 = 0 (._.  ._. = . because only Length is identical)
0  0 implies {} ( .  . implies {} 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/complement...irstaxioms.pdf
According to Monadic Mathematics the Number concept is the fundamental buildingblock of a nondestructive interaction between opposites, which is based on Includedmiddle 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/complement...aloisDialog.pdf
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Last edited by Doron Shadmi on 07272005 at 02:38 PM
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