Monadic Numbers (Monadic Calculus)

[Eodnhoj7]

(+3 → -2) → +1
(+3 → -2) ← +1
(+3 ← -2) → +1
(+3 ← -2) ← +1
(+3 ⇄ -2) → +1
(+3 ⇄ -2) ← +1
(+3 ⇄ -2) ⇄ +1

https://en.wikipedia.org/wiki/Reversible_computing




Ehh...as usual my answer is "yes" and "no" at the same time in different respects.

In the symbol grounding problem, we are left with the common denominator amidst all symbols as having a geometric origin premised in the point and extending from there due to the nature of all communication (and reality) having a nature of directed movement. The point, which is the origin for all symbols considering we put a pen on a piece of paper (or ancestors a stick in the dirt) and begin the symbol with a simple dot with the symbol gaining form and through the direction of that dot on the piece of paper or in the dirt.

The dot is the common foundation of all symbols, and its directed nature the secondary aspect through which the symbol exists with the symbol on its own terms being reduced again to a point of origin or dot from a literal or intellectual distance where it gains it's meaning by being directed to further symbols.




We can see in these following equations, where the numbers as both form and function are defined by their directional qualities, that a standard equation as a localized set of relations "generally" is directed towards one number as the answer. In these respects the equation is a cause and the answer is an effect. This effect, which I will observe further in the below, observes an inherent element of approximation in it considering the answer itself as an approximation of the equations observes an inherent element of deficiency as a thing in itself but rather an extension of it.

The equation as a cause can be observed in the symbol: ∙
Effect, or approximation, as: ":"


(+3 → -2)∙ → +1
(+3 ← -2)∙ → +1
(+3 ⇄ -2)∙ → +1


The same occurs in one respect for when a number is directed towards an equation, however considering all numbers as exist through an infinite series of both quantitative forms (the actual number in a standard interpretation) and functions (a perpetual series of addition, substraction, division, multiplication, roots and powers that maintains itself as a constant through the number) all numbers as directed towards a specific equation observe that equation as a localization of infinite series and hence is an approximate structural extension of it. In these respects the equation has an element of randomness to it.

I will use the symbol ":" for randomness, where randomness (using the definition of chaos theory) is merely observing an approximate or part of a larger series in this case, as an extension of that series, where this equation as not a thing in itself is strictly a connector of the series. All equations, as premised from a number are inherently random in these respects, with the answer as a number being a "cause".


(+3 → -2): ← +1∙
(+3 ← -2): ← +1∙
(+3 ⇄ -2): ← +1∙


The problem occurs, or maybe less of a problem and more of a paradox, in that while the equation may in itself be directed towards a number, and that number exists as a directional property in itself, this number exists as an extension of the equation and as an extension exists through the equation as the equation. So while ((+3 ← -2) → +1 ∧ (+3 → -2) → +1) may be directed towards +1, +1 as an extension of this equation is directed back towards the equation as ((+3 → -2) ← +1 ∧ (+3 ← -2) ← +1) considering the equation exists if and only if +1 is the answer and therefore +1 as existing determines the equation.

The problem occurs, in standard math, that while x equation may equal y, y equals and infinite series of equations resulting in number lines in themselves hence the standard interpretation of math is quite literally dependent upon directional qualities in defining it.

So if we are to look at:

(+3 ⇄ -2) ⇄ +1

as

((+3 ← -2) → +1 ∧ (+3 → -2) → +1) ∧ ((+3 → -2) ← +1 ∧ (+3 ← -2) ← +1)

we are left with:

(+3 ⇄ -2)◬ ⇄ +1◬ where "◬" is strictly a symbol observing both cause "∙" and effects as an extension of cause synonymous to approximation as randomness ":" rather than observing the symbol "∙:" which is used in further numbers.

as:

((+3 ← -2) → +1 ∧ (+3 → -2) → +1) ∧ ((+3 → -2)ε ← +1 ∧ (+3 ← -2)ε ← +1)

This is considering the equation is both a cause and approximation (as effect) while the answer proceeding from the equation is equivalent to a cause for the equation itself.

Now considering the numbers which composed the equations, follow the same form and function to the answers they lead to (in the respect each number is an observation of an infinite series of further quantities as forms and the proceeding functions which manifest these numbers) each number in the equation has a simultaneous nature of "cause" and "effect" as "◬":

(+3◬ ⇄ -2◬)◬ ⇄ +1◬

****The equation itself is a localization of the monad and hence exists as a cause/effect in itself as ◬.



Now the reason "+3", "-2" and "+1" are observed as having an inherent element of approximation inherent within them is considering they exist as a series and are composed of a series. Each number as an infinite series of not just numbers, but multiple inherent series of arithmetic functions where each number is multiple arithmetic functions in itself, exists as a monadic number "⊙" where any approximation of this monadic number (while a cause in itself as an extension of the monadic number) is equivalent to ◬ in respect that each function is a localized portion of the number and not the whole number in itself. All of the numbers as monads reference the premise, where the monad is both 1 infinite form(number)/function series.








But your use of ⇄ is ambiguous.


(+3 ⇄ -2) ⇄ +1


((+3 ← -2) → +1 ∧ (+3 → -2) → +1) ∧ ((+3 → -2) ← +1 ∧ (+3 ← -2) ← +1)




(+4 ⇄ -3) ⇄ +1

(+4 ← -3) → +1 ∧ No (-3 → -2) → +1 -5 not -1



(+4 ⇄ -3) ⇄ (+3 ⇄ -2)
+4 ⇄ -3 ⇄ +3 ⇄ -2
(+4 ⇄ -3 ⇄ +3 ⇄ -2)⇄+2

or more specifically:

(+4◬ ⇄ -3◬ ⇄ +3◬ ⇄ -2◬)◬ ⇄ +2◬




What?

[TimeSeeker]

(+3 → -2) → +1
(+3 → -2) ← +1
(+3 ← -2) → +1
(+3 ← -2) ← +1
(+3 ⇄ -2) → +1
(+3 ⇄ -2) ← +1
(+3 ⇄ -2) ⇄ +1

https://en.wikipedia.org/wiki/Reversible_computing




Ehh...as usual my answer is "yes" and "no" at the same time in different respects.

In the symbol grounding problem, we are left with the common denominator amidst all symbols as having a geometric origin premised in the point and extending from there due to the nature of all communication (and reality) having a nature of directed movement. The point, which is the origin for all symbols considering we put a pen on a piece of paper (or ancestors a stick in the dirt) and begin the symbol with a simple dot with the symbol gaining form and through the direction of that dot on the piece of paper or in the dirt.

The dot is the common foundation of all symbols, and its directed nature the secondary aspect through which the symbol exists with the symbol on its own terms being reduced again to a point of origin or dot from a literal or intellectual distance where it gains it's meaning by being directed to further symbols.




We can see in these following equations, where the numbers as both form and function are defined by their directional qualities, that a standard equation as a localized set of relations "generally" is directed towards one number as the answer. In these respects the equation is a cause and the answer is an effect. This effect, which I will observe further in the below, observes an inherent element of approximation in it considering the answer itself as an approximation of the equations observes an inherent element of deficiency as a thing in itself but rather an extension of it.

The equation as a cause can be observed in the symbol: ∙
Effect, or approximation, as: ":"


(+3 → -2)∙ → +1
(+3 ← -2)∙ → +1
(+3 ⇄ -2)∙ → +1


The same occurs in one respect for when a number is directed towards an equation, however considering all numbers as exist through an infinite series of both quantitative forms (the actual number in a standard interpretation) and functions (a perpetual series of addition, substraction, division, multiplication, roots and powers that maintains itself as a constant through the number) all numbers as directed towards a specific equation observe that equation as a localization of infinite series and hence is an approximate structural extension of it. In these respects the equation has an element of randomness to it.

I will use the symbol ":" for randomness, where randomness (using the definition of chaos theory) is merely observing an approximate or part of a larger series in this case, as an extension of that series, where this equation as not a thing in itself is strictly a connector of the series. All equations, as premised from a number are inherently random in these respects, with the answer as a number being a "cause".


(+3 → -2): ← +1∙
(+3 ← -2): ← +1∙
(+3 ⇄ -2): ← +1∙


The problem occurs, or maybe less of a problem and more of a paradox, in that while the equation may in itself be directed towards a number, and that number exists as a directional property in itself, this number exists as an extension of the equation and as an extension exists through the equation as the equation. So while ((+3 ← -2) → +1 ∧ (+3 → -2) → +1) may be directed towards +1, +1 as an extension of this equation is directed back towards the equation as ((+3 → -2) ← +1 ∧ (+3 ← -2) ← +1) considering the equation exists if and only if +1 is the answer and therefore +1 as existing determines the equation.

The problem occurs, in standard math, that while x equation may equal y, y equals and infinite series of equations resulting in number lines in themselves hence the standard interpretation of math is quite literally dependent upon directional qualities in defining it.

So if we are to look at:

(+3 ⇄ -2) ⇄ +1

as

((+3 ← -2) → +1 ∧ (+3 → -2) → +1) ∧ ((+3 → -2) ← +1 ∧ (+3 ← -2) ← +1)

we are left with:

(+3 ⇄ -2)◬ ⇄ +1◬ where "◬" is strictly a symbol observing both cause "∙" and effects as an extension of cause synonymous to approximation as randomness ":" rather than observing the symbol "∙:" which is used in further numbers.

as:

((+3 ← -2) → +1 ∧ (+3 → -2) → +1) ∧ ((+3 → -2)ε ← +1 ∧ (+3 ← -2)ε ← +1)

This is considering the equation is both a cause and approximation (as effect) while the answer proceeding from the equation is equivalent to a cause for the equation itself.

Now considering the numbers which composed the equations, follow the same form and function to the answers they lead to (in the respect each number is an observation of an infinite series of further quantities as forms and the proceeding functions which manifest these numbers) each number in the equation has a simultaneous nature of "cause" and "effect" as "◬":

(+3◬ ⇄ -2◬)◬ ⇄ +1◬

****The equation itself is a localization of the monad and hence exists as a cause/effect in itself as ◬.



Now the reason "+3", "-2" and "+1" are observed as having an inherent element of approximation inherent within them is considering they exist as a series and are composed of a series. Each number as an infinite series of not just numbers, but multiple inherent series of arithmetic functions where each number is multiple arithmetic functions in itself, exists as a monadic number "⊙" where any approximation of this monadic number (while a cause in itself as an extension of the monadic number) is equivalent to ◬ in respect that each function is a localized portion of the number and not the whole number in itself. All of the numbers as monads reference the premise, where the monad is both 1 infinite form(number)/function series.








But your use of ⇄ is ambiguous.


(+3 ⇄ -2) ⇄ +1


((+3 ← -2) → +1 ∧ (+3 → -2) → +1) ∧ ((+3 → -2) ← +1 ∧ (+3 ← -2) ← +1)




(+4 ⇄ -3) ⇄ +1

(+4 ← -3) → +1 ∧ No (-3 → -2) → +1 -5 not -1



(+4 ⇄ -3) ⇄ (+3 ⇄ -2)
+4 ⇄ -3 ⇄ +3 ⇄ -2
(+4 ⇄ -3 ⇄ +3 ⇄ -2)⇄+2

or more specifically:

(+4◬ ⇄ -3◬ ⇄ +3◬ ⇄ -2◬)◬ ⇄ +2◬




What?

All you have done is moved the goal post. ⇄ is no longer ambiguous. Because you have now hidden all the ambiguity behind a new symbol: ◬

What does ◬ mean ? What system/mechanism/function/operation/algorithm does it describe?

[Eodnhoj7]

https://en.wikipedia.org/wiki/Reversible_computing




Ehh...as usual my answer is "yes" and "no" at the same time in different respects.

In the symbol grounding problem, we are left with the common denominator amidst all symbols as having a geometric origin premised in the point and extending from there due to the nature of all communication (and reality) having a nature of directed movement. The point, which is the origin for all symbols considering we put a pen on a piece of paper (or ancestors a stick in the dirt) and begin the symbol with a simple dot with the symbol gaining form and through the direction of that dot on the piece of paper or in the dirt.

The dot is the common foundation of all symbols, and its directed nature the secondary aspect through which the symbol exists with the symbol on its own terms being reduced again to a point of origin or dot from a literal or intellectual distance where it gains it's meaning by being directed to further symbols.




We can see in these following equations, where the numbers as both form and function are defined by their directional qualities, that a standard equation as a localized set of relations "generally" is directed towards one number as the answer. In these respects the equation is a cause and the answer is an effect. This effect, which I will observe further in the below, observes an inherent element of approximation in it considering the answer itself as an approximation of the equations observes an inherent element of deficiency as a thing in itself but rather an extension of it.

The equation as a cause can be observed in the symbol: ∙
Effect, or approximation, as: ":"


(+3 → -2)∙ → +1
(+3 ← -2)∙ → +1
(+3 ⇄ -2)∙ → +1


The same occurs in one respect for when a number is directed towards an equation, however considering all numbers as exist through an infinite series of both quantitative forms (the actual number in a standard interpretation) and functions (a perpetual series of addition, substraction, division, multiplication, roots and powers that maintains itself as a constant through the number) all numbers as directed towards a specific equation observe that equation as a localization of infinite series and hence is an approximate structural extension of it. In these respects the equation has an element of randomness to it.

I will use the symbol ":" for randomness, where randomness (using the definition of chaos theory) is merely observing an approximate or part of a larger series in this case, as an extension of that series, where this equation as not a thing in itself is strictly a connector of the series. All equations, as premised from a number are inherently random in these respects, with the answer as a number being a "cause".


(+3 → -2): ← +1∙
(+3 ← -2): ← +1∙
(+3 ⇄ -2): ← +1∙


The problem occurs, or maybe less of a problem and more of a paradox, in that while the equation may in itself be directed towards a number, and that number exists as a directional property in itself, this number exists as an extension of the equation and as an extension exists through the equation as the equation. So while ((+3 ← -2) → +1 ∧ (+3 → -2) → +1) may be directed towards +1, +1 as an extension of this equation is directed back towards the equation as ((+3 → -2) ← +1 ∧ (+3 ← -2) ← +1) considering the equation exists if and only if +1 is the answer and therefore +1 as existing determines the equation.

The problem occurs, in standard math, that while x equation may equal y, y equals and infinite series of equations resulting in number lines in themselves hence the standard interpretation of math is quite literally dependent upon directional qualities in defining it.

So if we are to look at:

(+3 ⇄ -2) ⇄ +1

as

((+3 ← -2) → +1 ∧ (+3 → -2) → +1) ∧ ((+3 → -2) ← +1 ∧ (+3 ← -2) ← +1)

we are left with:

(+3 ⇄ -2)◬ ⇄ +1◬ where "◬" is strictly a symbol observing both cause "∙" and effects as an extension of cause synonymous to approximation as randomness ":" rather than observing the symbol "∙:" which is used in further numbers.

as:

((+3 ← -2) → +1 ∧ (+3 → -2) → +1) ∧ ((+3 → -2)ε ← +1 ∧ (+3 ← -2)ε ← +1)

This is considering the equation is both a cause and approximation (as effect) while the answer proceeding from the equation is equivalent to a cause for the equation itself.

Now considering the numbers which composed the equations, follow the same form and function to the answers they lead to (in the respect each number is an observation of an infinite series of further quantities as forms and the proceeding functions which manifest these numbers) each number in the equation has a simultaneous nature of "cause" and "effect" as "◬":

(+3◬ ⇄ -2◬)◬ ⇄ +1◬

****The equation itself is a localization of the monad and hence exists as a cause/effect in itself as ◬.



Now the reason "+3", "-2" and "+1" are observed as having an inherent element of approximation inherent within them is considering they exist as a series and are composed of a series. Each number as an infinite series of not just numbers, but multiple inherent series of arithmetic functions where each number is multiple arithmetic functions in itself, exists as a monadic number "⊙" where any approximation of this monadic number (while a cause in itself as an extension of the monadic number) is equivalent to ◬ in respect that each function is a localized portion of the number and not the whole number in itself. All of the numbers as monads reference the premise, where the monad is both 1 infinite form(number)/function series.








But your use of ⇄ is ambiguous.


(+3 ⇄ -2) ⇄ +1


((+3 ← -2) → +1 ∧ (+3 → -2) → +1) ∧ ((+3 → -2) ← +1 ∧ (+3 ← -2) ← +1)




(+4 ⇄ -3) ⇄ +1

(+4 ← -3) → +1 ∧ No (-3 → -2) → +1 -5 not -1



(+4 ⇄ -3) ⇄ (+3 ⇄ -2)
+4 ⇄ -3 ⇄ +3 ⇄ -2
(+4 ⇄ -3 ⇄ +3 ⇄ -2)⇄+2

or more specifically:

(+4◬ ⇄ -3◬ ⇄ +3◬ ⇄ -2◬)◬ ⇄ +2◬




What?

All you have done is moved the goal post. ⇄ is no longer ambiguous. Because you have now hidden all the ambiguity behind a new symbol: ◬

What does ◬ mean ? What system/mechanism/function/operation/algorithm does it describe?

1) Actually that is the issue in language: ambiguity. This is considering one symbol leads progress to another symbol with infinite variation. This infinite variation observe the symbol as having a generative quality that is synthetic. 1 symbol leads to another as 2 symbols with the totality of these symbols, the original and the original leading to the two symbols as approximates of the original symbols as a singularity leading to a dualism.

All generation, due to time, has a triadic nature hence all symbols are points of generation and this applies to math and logic as well.

Hence with the increase in precision through an atomization comes an increase in ambiguity as a unified form is reduced to multiple forms. This unified symbol


2) "◬" observes all numbers (and symbols for that matter when applied to language) are both causes in themselves for further numbers and approximations as well of prior and future numbers.


Considering all numbers are monadic and extensions of the monad with all numbers being composed of an infinite series of further numbers and an infinite series of further functions, the monadic number is a complete number as each monadic number contains all numbers in it as well as all arithmetic functions as the base for further mathematics.


The monadic numbers each contain all numbers in them through an infinite progression and series of functions where the progressive nature of the number through the progressive function observes all numbers as not just an infinite series but an infinite series of an infinite series where all numbers are extensions of eachother in each number. In theory each number is composed of all equations where any standard approximation of the monadic number is strictly an observation of the equation inherent within it where the equation itself is fundamentally random as a localization of certain series within the number.


Monadic numbers exist through simpler series as:



1⊙= 1+1-1, 1+2-2, 1+3-3...
1*1, 1*2-1/1...
...

2⊙= 1+1, 1+2-1, 1+3-2...
2+1-1, 2+2-2, 2+3-3...
3-1, 3-2+1, 3-3+2....
...

(1/1)+(1/1), (1/1)+(2/2), (1/1)+(3/3)...
(1*1)+(1*1), (1*1)+(2*1)-(1*1), (1*1)+(3*1)-(2*1)...
...

or more complex series:

1⊙= (1+(7/7)/(1000/1000))-1*1, (1+(8/8)/(1001/1001))-1*(1+1-1), (1+(9/9)/(1002/1002))-1*(1+2-2)....

Where any equation such as 2+3=5 is an approximation of all monadic numbers and has an inherent element of randomness in it because of this:

1⊙= (2+3=5)-4, (2+3=5)-5+6, (2+3=5)-7+3...
...

3⊙= (2+3=5)-2, (2+3=5)-3+1, (2+3=5)-4+2...
...

111⊙= (2+3=5)+106, (2+3=5)+107-1, (2+3=5)+108-2...
...

1005⊙= (2+3=5)+1000, (2+3=5)+1001-1, (2+3=5)+1002-2...
...

And considering all equations are determined by there directional properties all equations, as well as the numbers inherent within them, maintain a dual nature of cause and effect.

Hence (+2◬→+3◬)◬→+5◬




"◬" is a symbol for the approximation of the monad by observing the number, or equation, as representing only a part in one respect while being subject to the same form and function; hence it is still a cause. The directional nature of math and language observes an inherent cause/effect paradigm where the number/word because of its directed cause to a further cause as effect (considering the number the equation results in is an extension of a monadic number as well as the numbers which form the equation) is in itself always correct and always random.

[gaffo]

Monadic Numbers

1) All numbers exist through continuums. 1 exists through 2,3,4..., with each number respectively existing through further numbers as a continuum of continuum.

A. This continuous nature of number necessitates not just a number line, but effectively the number as having a directional property.

Example:

1 -> 2 -> 3 -> 4 -> 5....

This nature of all numbers as progressive in nature as defined by thier directional qualities observes all numbers as linear directed movement.

B. The continuous nature of number observes and inherent circulatory nature through the number line where all number cycle to form the number line as progressive oscillation.

Example:

(1 -> 1) -> 2

(1 <- 2) -> 3

(2 -> 2) -> 4

...

The nature of number as 1 directed through itself to maintain itself as 1 through 2, with 2 as an extension of 1 directed towards itself as 4 and 2 directed back to 1 as 3, observes that 1 maintains itself as absolute while progressing through a continuum which observes the same qualities.

C. All numbers, through 1 as extensions of 1, are points of oscillation where this point of oscillation observes the number directed through itself as itself through an approximate number which this approximate number being 1 observed through variations of one. These variations occur as 1 directed mirroring 0 as absent of structure. 1 mirroring 0, with 0 as void of meaning, observes 1 replicating this formless nature through multiple variations with 0 as void being a point of inversion between unity and multiple units.

1 exists as a point of origin existing through itself as itself.

All numbers as an origin point(s) through point(s) as points(s) observe all monastic numbers as infinite Continuums of numbers.



2) All numbers exist through continuums as continuous functions with the number line as a continuum observing all functions as continuums.

a. 1= 1+1-1, 1+2-2, 1+3-3, 1+4-4, 1+5-5...
b. 1= 2-1, 3-2, 4-3, 5-4, 6-5, 7-6...
c. 1= 1*1/1, 1*2/2, 1*3/3, 1*4/4, 1*5/5...
d. 1= 1^1/1, 1^2/2, 1^3/3, 1^4/4, 1^5/5...
f. 1= 2root1, 2root2/2, 2root3/3, 2root4/4...

With infinite variation all showing all number exists as continuums through there inherent functions. While the numbers progressively vary within the continuum, the functions maintain themselves as constants as well as all number being extensions of one.

The number as inseparable from the function, as well as all number existing as an extension of one while being one continuum in itself (ex: 2 -> 4 -> 6...), observes 1 as a continuous function. All numbers are absolute as all numbers are continuums.

3. The number as composed of continuums, with these continuums existing through arithmetic functions as continuums, observes not just the number as a continuum but the function as a continuum as well where the number as form cannot be seperated from it as a function.

4. All numbers are inseperable from all functions. Hence +1 is inseperable from addition, multiplication and powers. -1 is inseperable from subtraction, division and root. One, as point 2 observes, is a continuous function as the standard arithmetic functions are continuous through one.

These arithmetic functions, as inseparable from all numbers through the 1, in themselves oscillate to form eachother in a similar manner to the number line.

Addition oscillate through itself as itself while manifesting multiplication as "the addition of addition".

Multiplication follows the same form and function where it maintains both addition and multiplication manifesting powers as the multiplication of multiplication.

Powers maintains itself as itself through addition and multiplication in turn acts a foundation for addition and multiplication.

Addition, multiplication and powers oscillate through eachother as eachother where 1 exists as an approximation of the other.

Negative functions (subtraction, division and root) follow this same oscillatory pattern.

5. All numbers as form/functions, must exist through further numbers as form/functions, hence all number inherently are directed through themselves as themselves where any replication is strictly an extension of the original mirror "numbers" as a replication of them. The number and the structural extension of that number through further numbers observe all numbers as mirroring as fundamentally oscillcations, progressing to further oscillations as an oscillation.

6. example:

1⊙= the number of oscillation of (2⊙,3⊙)
⧂ = The oscillation of the numbers into further oscillations.

1⊙(2⊙,3⊙) ⧂ (1⊙,2⊙,3⊙,4⊙,5⊙,6⊙,8⊙,9⊙,27⊙)

All numbers as a directive nature observes the directional qualities of the numbers forming new numbers under certain circumstances.
a. 2+3= 5
b. 3+2= 5
c. 2*3 = 6
d. 3*2 = 6
e. 2^3 = 8
f. 3^2 = 9
g. 2+2 = 4
h. 2*2 = 4
i. 2^2 = 4
j. 3+3 = 6
k. 3*3 = 9
l. 3^3 = 27

These mirror number in turn form further relations relative to other mirror numbers where all numbers exist through all numbers an each number is an extension of 1⊙. The monadic number as an oscillation of itself through further monadic numbers as oscillations observes an inherent repitition of monadic numbers, through monadic numbers as simultaneous extensions of the monadic numbers and acting as self referencing in accords to the monadic numbers they exist through.

All monadic calculus is a proof in itself as the repitition of monadic numbers forming an inherent symmetry resulting in both the statement and the numbers as self referencing absolutes as proofs in themselves.

Each monadic number, as a form/function, cycles to further proofs where all calculations are approximations of the Monad and have an inherent element of randomness. This randomness, where the repitition of structural extensions of the Monad are approximations of it, observes all proofs as both random and having self maintained structure reflecting this randomness through multiple monadic numbers as extensions of eachother and 1.

Each monadic number as a set of infinite numbers observe through a continuum, is in itself a continuum where one monadic number directed towards another is a set of infinite fractals in one respect while simultaneously results in a set of monadic numbers.

nice sophistry/word salad.

where do imaginary salads - i mean numbers, fit into your salad?

[TimeSeeker]

nice sophistry/word salad.

where do imaginary salads - i mean numbers, fit into your salad?

They fit exactly into the box they belong: useful mathematical tools with useful properties for manipulating complex numbers.

Well disconnected from reality, but useful in practice none the less.

[Eodnhoj7]

Monadic Numbers

1) All numbers exist through continuums. 1 exists through 2,3,4..., with each number respectively existing through further numbers as a continuum of continuum.

A. This continuous nature of number necessitates not just a number line, but effectively the number as having a directional property.

Example:

1 -> 2 -> 3 -> 4 -> 5....

This nature of all numbers as progressive in nature as defined by thier directional qualities observes all numbers as linear directed movement.

B. The continuous nature of number observes and inherent circulatory nature through the number line where all number cycle to form the number line as progressive oscillation.

Example:

(1 -> 1) -> 2

(1 <- 2) -> 3

(2 -> 2) -> 4

...

The nature of number as 1 directed through itself to maintain itself as 1 through 2, with 2 as an extension of 1 directed towards itself as 4 and 2 directed back to 1 as 3, observes that 1 maintains itself as absolute while progressing through a continuum which observes the same qualities.

C. All numbers, through 1 as extensions of 1, are points of oscillation where this point of oscillation observes the number directed through itself as itself through an approximate number which this approximate number being 1 observed through variations of one. These variations occur as 1 directed mirroring 0 as absent of structure. 1 mirroring 0, with 0 as void of meaning, observes 1 replicating this formless nature through multiple variations with 0 as void being a point of inversion between unity and multiple units.

1 exists as a point of origin existing through itself as itself.

All numbers as an origin point(s) through point(s) as points(s) observe all monastic numbers as infinite Continuums of numbers.



2) All numbers exist through continuums as continuous functions with the number line as a continuum observing all functions as continuums.

a. 1= 1+1-1, 1+2-2, 1+3-3, 1+4-4, 1+5-5...
b. 1= 2-1, 3-2, 4-3, 5-4, 6-5, 7-6...
c. 1= 1*1/1, 1*2/2, 1*3/3, 1*4/4, 1*5/5...
d. 1= 1^1/1, 1^2/2, 1^3/3, 1^4/4, 1^5/5...
f. 1= 2root1, 2root2/2, 2root3/3, 2root4/4...

With infinite variation all showing all number exists as continuums through there inherent functions. While the numbers progressively vary within the continuum, the functions maintain themselves as constants as well as all number being extensions of one.

The number as inseparable from the function, as well as all number existing as an extension of one while being one continuum in itself (ex: 2 -> 4 -> 6...), observes 1 as a continuous function. All numbers are absolute as all numbers are continuums.

3. The number as composed of continuums, with these continuums existing through arithmetic functions as continuums, observes not just the number as a continuum but the function as a continuum as well where the number as form cannot be seperated from it as a function.

4. All numbers are inseperable from all functions. Hence +1 is inseperable from addition, multiplication and powers. -1 is inseperable from subtraction, division and root. One, as point 2 observes, is a continuous function as the standard arithmetic functions are continuous through one.

These arithmetic functions, as inseparable from all numbers through the 1, in themselves oscillate to form eachother in a similar manner to the number line.

Addition oscillate through itself as itself while manifesting multiplication as "the addition of addition".

Multiplication follows the same form and function where it maintains both addition and multiplication manifesting powers as the multiplication of multiplication.

Powers maintains itself as itself through addition and multiplication in turn acts a foundation for addition and multiplication.

Addition, multiplication and powers oscillate through eachother as eachother where 1 exists as an approximation of the other.

Negative functions (subtraction, division and root) follow this same oscillatory pattern.

5. All numbers as form/functions, must exist through further numbers as form/functions, hence all number inherently are directed through themselves as themselves where any replication is strictly an extension of the original mirror "numbers" as a replication of them. The number and the structural extension of that number through further numbers observe all numbers as mirroring as fundamentally oscillcations, progressing to further oscillations as an oscillation.

6. example:

1⊙= the number of oscillation of (2⊙,3⊙)
⧂ = The oscillation of the numbers into further oscillations.

1⊙(2⊙,3⊙) ⧂ (1⊙,2⊙,3⊙,4⊙,5⊙,6⊙,8⊙,9⊙,27⊙)

All numbers as a directive nature observes the directional qualities of the numbers forming new numbers under certain circumstances.
a. 2+3= 5
b. 3+2= 5
c. 2*3 = 6
d. 3*2 = 6
e. 2^3 = 8
f. 3^2 = 9
g. 2+2 = 4
h. 2*2 = 4
i. 2^2 = 4
j. 3+3 = 6
k. 3*3 = 9
l. 3^3 = 27

These mirror number in turn form further relations relative to other mirror numbers where all numbers exist through all numbers an each number is an extension of 1⊙. The monadic number as an oscillation of itself through further monadic numbers as oscillations observes an inherent repitition of monadic numbers, through monadic numbers as simultaneous extensions of the monadic numbers and acting as self referencing in accords to the monadic numbers they exist through.

All monadic calculus is a proof in itself as the repitition of monadic numbers forming an inherent symmetry resulting in both the statement and the numbers as self referencing absolutes as proofs in themselves.

Each monadic number, as a form/function, cycles to further proofs where all calculations are approximations of the Monad and have an inherent element of randomness. This randomness, where the repitition of structural extensions of the Monad are approximations of it, observes all proofs as both random and having self maintained structure reflecting this randomness through multiple monadic numbers as extensions of eachother and 1.

Each monadic number as a set of infinite numbers observe through a continuum, is in itself a continuum where one monadic number directed towards another is a set of infinite fractals in one respect while simultaneously results in a set of monadic numbers.

nice sophistry/word salad.

where do imaginary salads - i mean numbers, fit into your salad?

No different than the rest, but considering the monadic number is primarily both a form and function through its nature as an infinite series of numbers.

The imaginary number i is defined solely by the property that its square is −1:
i 2 = − 1. {\displaystyle i^{2}=-1.} {\displaystyle i^{2}=-1.}
With i defined this way, it follows directly from algebra that i and −i are both square roots of −1.

https://en.wikipedia.org/wiki/Imaginary_unit


1⊙= (i^2= -1)+2, (i^6= -1)+2, (i^10= -1)+2...
(i^2= -1)+2-1+1, (i^6= -1)+2-2+2, (i^10= -1)+2-3+3...
...

i⊙= i^1,i^5,i^9...






The real question, relative to the monadic numbers is considering their inherent nature as arithmetic functions if a positive or negative monadic number has any different function in itself.

The statement:

1⊙(2⊙,3⊙) ⧂ (1⊙,2⊙,3⊙,4⊙,5⊙,6⊙,8⊙,9⊙,27⊙)

Is premised strictly in positive form and function:

a. 2+3= 5
b. 3+2= 5
c. 2*3 = 6
d. 3*2 = 6
e. 2^3 = 8
f. 3^2 = 9
g. 2+2 = 4
h. 2*2 = 4
i. 2^2 = 4
j. 3+3 = 6
k. 3*3 = 9
l. 3^3 = 27

But leads to further questions if each number is a quantitative series and function:


If it is premised in a strictly negative form and function:

1⊙(-2⊙,-3⊙) ⧂ (-6⊙, -5.19615242i⊙, -5⊙,-4.24264069i⊙, -4⊙, -3⊙, -2⊙, -1.73205081i⊙, -1.41421356i⊙,-1 1/2⊙, -1⊙, 2/3⊙, 1⊙, 1 1/2⊙, 1.41421356i⊙, 1.73205081i⊙, 4.24264069i⊙, 5.19615242i⊙)

a. -2-3= -5
b. -3-2= -5

c. -2/3 = -2/3
c1. -2/-3 = 2/3

d. -3/2 = -1 1/2
d1. -3/-2 = 1 1/2

e. 2root-3 = 1.73205081i
e1. -2root-3 = -1.73205081i

f. 3root-2 = 4.24264069i
f1. -3root-2 = -4.24264069i

g. -2-2 = -4

h. -2/2 = -1
h1. -2/-2 = 1

i. 2root-2 = 1.41421356i
i1. -2root-2 = -1.41421356i


j. -3-3 = -6

k. -3/3 = -1
k1. -3/-3 = 1

l. 3root-3 = 5.19615242i
l1. -3root-3 = -5.19615242i




It even extends further considering 1⊙ of 1⊙ (2⊙,3⊙) ⧂ (1⊙,2⊙,3⊙,4⊙,5⊙,6⊙,8⊙,9⊙,27⊙) observes the number of cycles:

a. 2+3= 5
b. 3+2= 5
c. 2*3 = 6
d. 3*2 = 6
e. 2^3 = 8
f. 3^2 = 9
g. 2+2 = 4
h. 2*2 = 4
i. 2^2 = 4
j. 3+3 = 6
k. 3*3 = 9
l. 3^3 = 27

and changes when: 2⊙(2⊙,3⊙) ⧂ (1⊙,2⊙,3⊙,4⊙,5⊙,6⊙,8⊙,9⊙,27⊙)
(8⊙, 10⊙, 12⊙, 16⊙, 18⊙, 25⊙, 36⊙, 64⊙, 81⊙, 256⊙, 729⊙, 3125⊙, 46,656⊙, 16,777,216⊙, 387,420,489⊙, 27^27⊙)

****Considering all monadic functions are dependent upon the replication of cycles, The oscillation of the numbers into further oscillations always contains the premise within the answer with the premise and answer always containing 1⊙ as the number of cycles replicating into further cycles is always 1 cycle containing further cycles. These oscillations observe an inherent arithmetic function nature involved.

Hence while:

1⊙(2⊙,3⊙) ⧂ (1⊙,2⊙,3⊙,4⊙,5⊙,6⊙,8⊙,9⊙,27⊙)

2⊙(2⊙,3⊙) ⧂ (8⊙, 10⊙, 12⊙, 16⊙, 18⊙, 25⊙, 36⊙, 64⊙, 81⊙, 256⊙, 729⊙, 3125⊙, 46,656⊙, 16,777,216⊙, 387,420,489⊙, 27^27⊙)


a. (2+3= 5 + 2+3= 5) = 10
(2+3= 5 * 2+3= 5) = 25
(2+3= 5 ^ 2+3= 5) = 3125


b. (3+2= 5 + 3+2= 5) = 10
(3+2= 5 * 3+2= 5) = 25
(3+2= 5 ^ 3+2= 5) = 3125

c. (2*3 = 6 + 2*3 = 6) = 12
(2*3 = 6 * 2*3 = 6) = 36
(2*3 = 6 ^ 2*3 = 6) = 46,656


d. (3*2 = 6 + 3*2 = 6) = 12
(3*2 = 6 * 3*2 = 6) = 36
(3*2 = 6 ^ 3*2 = 6) = 46,656

e. (2^3 = 8 + 2^3 = = 16
(2^3 = 8 * 2^3 = = 64
(2^3 = 8 ^ 2^3 = = 16,777,216


f. (3^2 = 9 + 3^2 = 9) = 18
(3^2 = 9 * 3^2 = 9) = 81
(3^2 = 9 ^ 3^2 = 9) = 387,420,489


g. (2+2 = 4 + 2+2=4) = 8
(2+2 = 4 * 2+2=4) = 16
(2+2 = 4 ^ 2+2=4) = 256




h. (2*2 = 4 + 2*2 = 4) = 8
(2*2 = 4 * 2*2 = 4) = 16
(2*2 = 4 ^ 2*2 = 4) = 256

i. (2^2 = 4 + 2^2 = 4) = 8
(2^2 = 4 * 2^2 = 4) = 16
(2^2 = 4 ^ 2^2 = 4) = 256

j. (3+3 = 6 + 3+3 = 6) = 12
(3+3 = 6 * 3+3 = 6) = 36
(3+3 = 6 ^ 3+3 = 6) = 46656

k. (3*3 = 9 + 3*3 = 9) = 18
(3*3 = 9 * 3*3 = 9) = 81
(3*3 = 9 ^ 3*3 = 9) = 387,420,489




l. (3^3 = 27 + 3^3 = 27) = 54
(3^3 = 27 * 3^3 = 27) = 729
(3^3 = 27 ^ 3^3 = 27) = 27^27


This further occurs respectively where the monadic number is a negative as well.



Because the number of cycles contains as series of cycles within it, the Oscillator (2⊙ in this case) always contains the series of cycles prior to it in the answer. So if the oscillator is 33⊙, it would contain as an answer the oscillations of 1⊙, 2⊙, 3⊙...33⊙. However considering (2⊙,3⊙) in the above example is a structural extension, or rather localization of the oscillations inherent within the oscillations of 1⊙ and 2⊙ these are not viewed as series. In these respects it maybe view as:

1⊙(2⊙,3⊙)◬ ⧂ (1⊙,2⊙,3⊙,4⊙,5⊙,6⊙,8⊙,9⊙,27⊙)◬

and

2⊙(2⊙,3⊙)◬ ⧂

(1⊙, 2⊙, 3⊙, 4⊙, 5⊙, 6⊙, 8⊙, 9⊙, 27⊙, 8⊙, 10⊙, 12⊙, 16⊙, 18⊙, 25⊙, 36⊙, 64⊙, 81⊙, 256⊙, 729⊙, 3125⊙, 46,656⊙, 16,777,216⊙, 387,420,489⊙, 27^27⊙)◬

Where each monadic number in itself is true as both an infinite series and function but as a set it is still an approximation. In these respects all monadic calculus through monadic numbers exists dually with a "triadic" number and "triadic function" (approximation of a whole conducive to randomness observed in the localization of number through any standard equation). The Monadic form/function exists dually with the Triadic form/function with this dualistic tension between the Monadic and Triadic observing a third neutral component where the premise and conclusion is the answer itself.

In these respects all monadic "equations" are localizations of self-referencing numbers and from a base level are best limited to 1 cycle. This is considering that because of the complexity, the majority of calculation would require a computer program as 1 simple equation of
33⊙(27⊙,23⊙) ⧂ x may be equivalent to pages upon pages of answers where the proof itself is strictly the book as self-referencing equation.



The real philosophical problem occurs considering all monadic numbers are composed of simultaneous positive and negative functions of arithmetic, as positive and negative number lines, the monadic number in itself cannot be strictly addition/multiplication/powers or subtraction/division/roots
but effectively both.

So the interpretation of 1⊙(2⊙,3⊙) ⧂ (1⊙,2⊙,3⊙,4⊙,5⊙,6⊙,8⊙,9⊙,27⊙) changes to:


when observing all monadic numbers as either positive or negative having inherent positive and negative values within them considering all of them exist through a constant series of functions.

results in:

*******Unfinished, out of time.


a. 2+3= 5
b. 2-3= -1
c. -2-3= -5

c. 3+2= 5
d. 3-2 = 1
e. -3-2=-5


e. 2*3 = 6
x. -2*3 = -6
x. 2*-3 = -6
x. -2*-3= 6
f. 2/3 = 2/3
g. -2/-3 = 2/3
x. -2/3 = -2/3
x. 2/-3 = -2/3


d. 3*2 = 6
x. -3*2 = -6
x. 3*-2 = -6
x. -3*-2 = 6
x. 3/2 = 3/2
x. -3/2 = -3/2
x. 3/-2 = -3/2
x. -3/-2 = 3/2

e. 2^3 = 8
x. 2root3=

x. 2^-3= 0.125
x. 2root-3=

x. -2^3= -8
x. -2root3=

x. -2^-3= -0.125
x. -2root-3


f. 3^2 = 9
x. 3root2=

g.-3^2 = 9
x.-3root2=


x. 3^-2 = 0.111111111
x. 3root-2=


x. -3^-2 = -0.111111111
x -3root-2=




g. 2+2 = 4
x. -2+2 = 0
x. 2+-2 = 0
x. -2-2 =4



h. 2*2 = 4
x. -2*2=-4
x. 2*-2=-4
x. -2*-2=4




i. 2^2 = 4
x. 2root2=

j. -2^2 = 4


x 2^-2


x. -2^-2




j. 3+3 = 6

k. 3*3 = 9

l. 3^3 = 27


1⊙(-2⊙,-3⊙) ⧂ (-6⊙, -5.19615242i⊙, -5⊙,-4.24264069i⊙, -4⊙, -3⊙, -2⊙, -1.73205081i⊙, -1.41421356i⊙,-1 1/2⊙, -1⊙, 2/3⊙, 1⊙, 1 1/2⊙, 1.41421356i⊙, 1.73205081i⊙, 4.24264069i⊙, 5.19615242i⊙)

a. -2-3= -5
b. -3-2= -5

c. -2/3 = -2/3
c1. -2/-3 = 2/3

d. -3/2 = -1 1/2
d1. -3/-2 = 1 1/2

e. 2root-3 = 1.73205081i
e1. -2root-3 = -1.73205081i

f. 3root-2 = 4.24264069i
f1. -3root-2 = -4.24264069i

g. -2-2 = -4

h. -2/2 = -1
h1. -2/-2 = 1

i. 2root-2 = 1.41421356i
i1. -2root-2 = -1.41421356i


j. -3-3 = -6

k. -3/3 = -1
k1. -3/-3 = 1

l. 3root-3 = 5.19615242i
l1. -3root-3 = -5.19615242i



The next question occurs if both positive and negative monadic numbers exist simultaneously through positive and negative functions is there really a difference between them?