Considering the confusion caused, I have figured it would be best to observe the Multidimensional Arithmetic thread form a seperate angle entirely, form a new set of notations and kill the old thread entirely in order to address the thread from a new angle which should, in theory, provide a greater degree of simplicity and efficiency in dealing with the problem of “space” at both the abstract and practical level along with its relation to math and logic.
Causality as Limit through 1 as Directional.
The mirroring function is premised on all numbers having directional qualities where these directive qualities define the number as a quantitative limit in itself. This directional nature of the mirroring function observes that numbers exist through a replication process that fundamentally is self-direction where a unit is directed into itself to form an approximate unit as an extension of this self-directed unit (with the premise of these self-directed units being quantitatively “1”).
We can see these directional qualities in the basic number line
1→2→3→4→5→∞
Or in the linear form of an equation:
(1+2=3) ∝ (+1→+2)=3
Or the simple fact that all empirically quantifiable entities are directed through time in one direction, hence quantification is grounded in linear 1 dimensional time with empiricism as the premise.
This mirroring function will be observed as the symbol: "⊙"
And the resulting numbers set as “mirrored in structure” or “replicated” will be observed under the symbol “⧂”
Considering all numbers, through the mirroring process are directed to themselves (and towards each other through themselves considering all numbers are extensions of 1) all numbers have an inherent directional quality of cycling observed as (⇄) which is equivalent symbolically in this use to ⟲. In these respects the number as an observation of direction, through the premises of quantities being 1 dimensional linear directions through time, maintain themselves as constants through an intradimensional nature where this “self-referencing” as “self-direction” maintains them as constants through an infinite cycle which is self-maintained and symmetrical.
Direction is the limit through which a quantity exists with the 1 extra-dimensional linear quality observing our standard definition of number as a finite quality which exists as units through time which relate. As intradimensional, all number exists through 1 as “unity” where any percieved multiplicity is an approximation of 1 as all numbers are extensions of 1 through 1 under an infinite cycle as 1 conducive to a geometric form of a circle. In these respects the mirror function is premised in the mirroring process of pythagoreanism of 1 where 1 is conducive to a 1 dimensional self-maintained point (not 0d).
This self-directive quality acts as a form of unification in itself and hence can be observed as “and” where “A and B” can be observed as “C” with “C” being observed as “and” in itself existing as the positive/neutral foundation of a unified relationship between variable in which it projects in itself as its own variable. Considering all numbers are inherently positive or negative they have an inherent nature of additive or subtractive which the form of the number and its function are inseparable, however assumed in all equations in the respect to how positive and negative number relate through addition and subtraction. These additive numbers as “and” or “converging” in nature have an inherent element of directionality where these subtractive numbers (as deficient in positive form and function) are separators and observe an “or” or “diverging” nature (where “A converging with B” observes a unity in direction and “A diverging from B” observes an inherent degree of seperation where one seperates the other into a new direction).
In these respects all numbers have inherent functional qualities inseperable from the number where:
(x) = +x
((x)) = *x
(((x))) = ^x
and inversely negatives are observed as:
)x( = -x
))x(( = /x
)))x((( = x root
The functions are inseparable from the form of the number as the form exists through the function and the function exists through the form. All alternation between the two phenomena of form and function observes an inherent element of directionality as a triadic element which is the neutral median from which both extend. Form as constant, observes direction fundamentally as a limit, while function as change observes direction as no-limit.
The nature of limit exists through no-limit as the limit itself must be constant and unchanging with any change being the application of new limit causing the previous limit to become finite. The nature of no-limit provides the foundation for limit as no-limit cancels itself out under its own terms, as a double negation, which maintains the positive “limit” considering “no-limit” observes “no” through “limit” and not a thing in itself but rather an observation of deficiency.
The mirror function observes an approximation of a unified reflective act in degrees of repitition where while the mirroring process is both one and infinite as all numbers follow this function simultaneously as 1, the function observes the inherent degrees of intradimensional repititition as an approximation of this infinity itself through an inherent multiplicity that exists as an extension of it. For example, all numbers exist simultaneously through 1, however one cannot view all numbers simultaneously. However the numbers are present nonetheless as constants through 1, hence any application of the mirroring of the intradimensional nature of the mirroring process, as one, is in itself quantifiable as 1 direction as all direction with all numbers extending from 1 observing multiple degrees of the process itself, hence it can be calculated through a multitude of degrees.
All resulting numbers are extensions of the numbers being mirrored.
Because of the complexity of the increase in degrees and numbers applied to the function, theoretically speaking it would be more efficient to apply these calculations through a potential computer program.
Premise examples:
1⊙(1) ⧂ ((1,2))
a) (1)⟲ =
((1)→(1)) ∧((1)←(1))=
((1⇄1)) =
((2)) ∋(2),((1)),(1)
2⊙(1) ⧂ ((((1,2,4))))
a) (1)⟲ =
((((1)→(1)) ∧((1)←(1))) → (((1)→(1)) ∧((1)←(1)))) ∧ ((((1)→(1)) ∧((1)←(1))) ← (((1)→(1)) ∧((1)←(1)))) =
((((1⇄1⇄1⇄1)))) =((((4)))) ∋((((2)))),((((1))))*****
Example 1:
1⊙(1),(2) ⧂ ((1,2,3,4))
a) (1)⟲ = ((1)→(1)) ∧((1)←(1))=
((1⇄1)) =
((2)) ∋(2),((1)),(1)
b) (2)⟲ =
((2)→(2)) ∧((2)←(2))= ((2⇄2)) =
((4)) ∋(4),((2)),(2),((1)),(1)
c) (1),(2)⟲ =
((1)→(2)) ∧((1)←(2))=
((1⇄2)) =
((3)) ∋(3),((2)),(2),((1)),(1)
In this above example the numbers 1,2,3 and 4 exist inseperable from addition and multiplication and hence as additive and multiplicative numbers.
Example 2:
1⊙(3),(2) ⧂ (((1,2,3,4,6)))
a) ((3))⟲ =
(((3))→((3))) ∧(((3))←((3))) =
(((3⇄3))) =
(((9))) ∋((9)),(9),(((3))),((3)),(3),(((1))),((1)),(1)
with ((3)) equivalent to: ((1) ⇄ 1 ⇄ 1)
(((1))) ⇄ (1) ⇄ (1))
⇅
(((1))) ⇄ (1) ⇄ (1)) = (((9)))
⇅
(((1))) ⇄ (1) ⇄ (1))
a1) (3)⟲ =
((3⇄3)) =
((6)) ∋(6),((3)),(3),((1)),(1)
***Considering ((3)) contains as an element of (3), and thereby in effect is an extension of it, (3) is observed as self-reflecting as well.
b) ((2))⟲ =
(((2))→((2))) ∧(((2))←((2)))=
(((2⇄2))) =
(((4))) ∋((4)),(4),(((2))),((2)),(2),(((1))),((1)),(1)
b1) (2)⟲ =
((2⇄2)) =
((4)) ∋(4),((2)),(2),((1)),(1)
b) ((3)),((2))⟲ = (((3))→((2))) ∧(((3))←((2)))= (((3⇄2))) =
(((6))) ∋((6)),(6),(((3))),((3)),(3),(((2))),((2)),(2) ,(((1))),((1)),(1)
In the above example the numbers 1,2,3,4 and 6 are inseparable from addition, multiplication and powers; hence are additive, muliplicative and power numbers.
Example 3:
1⊙(1), )2( ⧂ ((1,2,)) ))1,2,4((
a) (1)⟲ = ((1)→(1)) ∧((1)←(1))= ((1⇄1)) =((2)) ∋(2),((1)),(1)
b) )2( ⟲ =
))2((⇄))2(( =
))4(( ∋ )4( ))2(( )2( ))1(( )1(
c) (1),)2( ⟲ =
((1)→ )2() ∧((1)←(2))=
((1)⇄ )2() =
)1( ∋ ((1)),(1),((1)),)2( )1(
In these respects the number line exists as a cycle that maintains itself while simultaneously expanding, hence maintains a dual circular and linear nature through the mirror function.
1→2 = 1⟲ →2
1→2→3→ 4 = (1,2)⟲ →3 and 2⟲ →4
1→2→3→4→5→6→∞ = (2,3)⟲ →5 and 3⟲ →6
Considering the mirroring process is constant and simultaneous for all number, 1 mirrors to infinity and cycles as such. In these respect “1 converges as infinity” exists as the foundation of the mirroring process where quantity as direction is founded in an intra dimensional self-referencing nature. Direction in turn exists as quantity since all number exist if and only if they are directed towards themselves through 1 with 1 existing as infinite direction. In these respects 1 as direction maintains a dualistic nature of “limit” and “no-limit” (with limit being direction as constant and unchanging). Direction as limit, in these regards provides the foundation of both quantity and quality.
(1⇄∞)
This nature of limit is causal in the respect it provides the further foundation for limit with this limit as effect being an approximate cause (as extension of original cause) and cause for further limits. In these respects, 1 as limit through direction observes a nature of cause and effect (approximate cause) where effect as approximation observes an inherent degree of randomness as a deficiency in structure respective of 1. Randomness, as deficiency, exists in turn as a negative dimension (synonymous to a negative number) as an approximator which observes a connection between existing 1 dimensional points in space which can only be observed through an inherent multiplicity.
This connection between multiple 1 dimensional points, through the -1 dimensional line (as an absence of direction which can only be observes through the self-directional nature of the 1 dimensional points) observes that all quantities are a series of connected points. In these respects a geometric quality, such as a triangle (3 1d points connected through 3 -1d lines) or square (4 1d points connected through 6 -1d lines), exists as a quantity in itself. So 3 points observes the triangle as a quantity, or 4 points the square/trapezoid/rectangle/etc. as a quantity in itself. In these respects all base geometric forms, as the foundation of all phenomena are in effect quantitative. These geometric forms as quantities in turn observe an infinite number of geometric shapes as grades which exist within the quantities themselves and hence quantity provides a foundation for quality while dually quality as direction provides a foundation for quantity.
This mirroring process in turn provides a foundation for not just quantitative math but qualitative logic as well considering a gradation of quantity results in quality as an extension of it through “the variable”.
Hence where addition observes an inherent unification as convergence as an extension of whole, multiplication/powers observes this unification as a localized finite version as convergence where convergence exists as a grade in itself resulting in “locality”. Hence the statement:
⊙(a),(b) ⧂ ((a,b,a_a,b_b,c,u))
a) (a)⟲ =
((a)→(a)) ∧((a)←(a)) =
((a⇄a)) =
((a_a )) ∋(a_a ),((a)),(a),(a_a ),((u)),(u)
*****(u) exists as universal
*****((p)) exists as particle/grade of universal which in itself is a universal
b) (b)⟲ =
((b)→(b)) ∧((b)←(b))=
((b⇄b)) =
((b_b )) ∋(b_b ),((b)),(b) (a_a ),((u)),(u)
c) (a),(b)⟲ =
((a)→(b)) ∧((a)←(b))=
((a⇄b)) =
((c=a∧b)) ∋(c=a∧b),((a)),(a),((b)),(b),((u)),(u)
Observes:
1) (a,b,a_a,b_b,c,u) as unified constants in themselves as extensions of “u” (universal) where a_a and b_b existing as grades of “a” and “b” respectively as universal constants in themselves.
2) ((a,b,a_a,b_b,c,u)) as multiples, or localities, which exist as parts or approximates of constants.
Mirror Function (MDR Update II)
Re: Mirror Function (MDR Update II)
Premise:
The form of a number is inseparable from its function considering 1 existing through an infinite continuum of arithmetic functions composing it. The arithmetic functions composing the number exist if and only if there is an infinite progression of numbers.
The number and function are inseparable, in these respects,
a. All numbers are sets of numbers, and because all numbers are functions all sets composed of functions. Sets are both number and function.
b. The number line is a continuous function.
c. Number as form and function necessitates a function that is also a quantity.
All numbers are intradimensional. 1 perpetually exists through itself as itself where all number is an extension of one.
1) Addition observes unification and its inherent within all positive numbers. Unity is founded in 1.
2) Subtraction observes seperation as approximation of all positive numbers considering all positive numbers are connected through 1. A negative number, as subtractive 1, is a connector as it occurs between numbers.
3) All numbers exist through there progression to further numbers, as evidenced by the basic number line. All numbers, through the number line as direction and the act of quantifying phenomenon as directed through time, are directional in nature
1 ↔ 2 ↔ 3 ↔ 4 ↔ 5 ↔ 6 ↔ 7 ↔ 8 ↔ 9 ↔ ...
1 ↔ 1 ∴ -1
1 ↔ 2 ∴ -1, -2, -3 (-2 as a connector between 2 and 1 with -1 as connector within for a total of -3 connectors containing as elements -1,-2,-3.
In these respects all numbers are formed through a progressive (linear) circularity where the progressive nature of the line allows for further number and the circle maintains the original number as a foundational element.
1) +1 → +2 ∋(+1,+1)
2) (+1 ← +2 ∋(+1,+1)) ∈ +3 ∵ +2 ∋ +1
3) (+2 ∋(+1,+1) → +2 ∋(+1,+1)) ∈ +4 ∵ +2 ∋ +1
(Steps 1,2,3 as 1) (+2 ∋(+1,+1) → +3 ∋(+1,+1,+1) → +4 ∋(+1,+1,+1,+1)) ∈ +9 ∵ (+2,+3,+4) ∋ +1
4) +1 ↔ ((+1 → +1)= (+1 ⇆ +1) = (+1⟲) = (+1 = ⟲))
5) ((+1 → +1)= (+1 ⇆ +1) = (+1⟲) = (+1 = ⟲)) = ⟨+1⟲|(+1 → +1)⟩ ∴ +1 ↔ ⟨"⟲"|"→")⟩
*** "(+1 → +1)" ∈ +1,+2 where "→" observers a continual fractal as continual infinite inversion:
1 → 1.1 → 1.2 → 1.3 → 1.4 → 1.5 → 1.6 → 1.7 → 1.8 → 1.9 → 2
1.1 → 1.11 → 1.12 → 1.13 → 1.14 → 1.15 → 1.16 → 1.17 → 1.18 → 1.19 → 1.2
1.11 → 1.111 → 1.112 → 1.113 → 1.114 → 1.115 → 1.116 → 1.117 → 1.118 → 1.119 → 1.12
1+(1/n→∞)
6) +1 = 0 where +1 = 0 are point space actthe foundations of quantity and quality. All "→" are an infinite series of fractals equivalent to infinite lines composing a line.
7) "(a)⊙" acts as the number of cycles the numbers are cycled through eachother considering all numbers are cycles:
((+1 → +1)= (+1 ⇆ +1) = (+1⟲) = (+1 = ⟲)) = ⟨+1⟲|(+1 → +1)⟩ ∴ +1 ↔ ⟨"⟲"|"→")⟩
This cyclical nature of number that is inseperable from number itself observes "x,y" in the statement "(a)⊙(x,y)" as cycles in themselves, hence the form/function results in further form functions where all statements exist as proofs in themselves. In these respects all numbers as form/functions are simultaneously sets where the form function exists as a set of numbers.
1⊙(1) ⧂ ((1,2))
a) (1)⟲ =
((1)→(1)) ∧((1)←(1))=
((1⇄1)) =
((2)) ∋(2),((1)),(1)
2⊙(1) ⧂ ((((1,2,4))))
a) (1)⟲ =
((((1)→(1)) ∧((1)←(1))) → (((1)→(1)) ∧((1)←(1)))) ∧ ((((1)→(1)) ∧((1)←(1))) ← (((1)→(1)) ∧((1)←(1)))) =
((((1⇄1⇄1⇄1)))) =((((4)))) ∋((((2)))),((((1))))*****
The form of a number is inseparable from its function considering 1 existing through an infinite continuum of arithmetic functions composing it. The arithmetic functions composing the number exist if and only if there is an infinite progression of numbers.
The number and function are inseparable, in these respects,
a. All numbers are sets of numbers, and because all numbers are functions all sets composed of functions. Sets are both number and function.
b. The number line is a continuous function.
c. Number as form and function necessitates a function that is also a quantity.
All numbers are intradimensional. 1 perpetually exists through itself as itself where all number is an extension of one.
1) Addition observes unification and its inherent within all positive numbers. Unity is founded in 1.
2) Subtraction observes seperation as approximation of all positive numbers considering all positive numbers are connected through 1. A negative number, as subtractive 1, is a connector as it occurs between numbers.
3) All numbers exist through there progression to further numbers, as evidenced by the basic number line. All numbers, through the number line as direction and the act of quantifying phenomenon as directed through time, are directional in nature
1 ↔ 2 ↔ 3 ↔ 4 ↔ 5 ↔ 6 ↔ 7 ↔ 8 ↔ 9 ↔ ...
1 ↔ 1 ∴ -1
1 ↔ 2 ∴ -1, -2, -3 (-2 as a connector between 2 and 1 with -1 as connector within for a total of -3 connectors containing as elements -1,-2,-3.
In these respects all numbers are formed through a progressive (linear) circularity where the progressive nature of the line allows for further number and the circle maintains the original number as a foundational element.
1) +1 → +2 ∋(+1,+1)
2) (+1 ← +2 ∋(+1,+1)) ∈ +3 ∵ +2 ∋ +1
3) (+2 ∋(+1,+1) → +2 ∋(+1,+1)) ∈ +4 ∵ +2 ∋ +1
(Steps 1,2,3 as 1) (+2 ∋(+1,+1) → +3 ∋(+1,+1,+1) → +4 ∋(+1,+1,+1,+1)) ∈ +9 ∵ (+2,+3,+4) ∋ +1
4) +1 ↔ ((+1 → +1)= (+1 ⇆ +1) = (+1⟲) = (+1 = ⟲))
5) ((+1 → +1)= (+1 ⇆ +1) = (+1⟲) = (+1 = ⟲)) = ⟨+1⟲|(+1 → +1)⟩ ∴ +1 ↔ ⟨"⟲"|"→")⟩
*** "(+1 → +1)" ∈ +1,+2 where "→" observers a continual fractal as continual infinite inversion:
1 → 1.1 → 1.2 → 1.3 → 1.4 → 1.5 → 1.6 → 1.7 → 1.8 → 1.9 → 2
1.1 → 1.11 → 1.12 → 1.13 → 1.14 → 1.15 → 1.16 → 1.17 → 1.18 → 1.19 → 1.2
1.11 → 1.111 → 1.112 → 1.113 → 1.114 → 1.115 → 1.116 → 1.117 → 1.118 → 1.119 → 1.12
1+(1/n→∞)
6) +1 = 0 where +1 = 0 are point space actthe foundations of quantity and quality. All "→" are an infinite series of fractals equivalent to infinite lines composing a line.
7) "(a)⊙" acts as the number of cycles the numbers are cycled through eachother considering all numbers are cycles:
((+1 → +1)= (+1 ⇆ +1) = (+1⟲) = (+1 = ⟲)) = ⟨+1⟲|(+1 → +1)⟩ ∴ +1 ↔ ⟨"⟲"|"→")⟩
This cyclical nature of number that is inseperable from number itself observes "x,y" in the statement "(a)⊙(x,y)" as cycles in themselves, hence the form/function results in further form functions where all statements exist as proofs in themselves. In these respects all numbers as form/functions are simultaneously sets where the form function exists as a set of numbers.
1⊙(1) ⧂ ((1,2))
a) (1)⟲ =
((1)→(1)) ∧((1)←(1))=
((1⇄1)) =
((2)) ∋(2),((1)),(1)
2⊙(1) ⧂ ((((1,2,4))))
a) (1)⟲ =
((((1)→(1)) ∧((1)←(1))) → (((1)→(1)) ∧((1)←(1)))) ∧ ((((1)→(1)) ∧((1)←(1))) ← (((1)→(1)) ∧((1)←(1)))) =
((((1⇄1⇄1⇄1)))) =((((4)))) ∋((((2)))),((((1))))*****