This Arithmetic will be called “Multi-Dimensional Reflective Arithmetic” or M.D.R Arithmetic for short.

M.D.R Arithmetic is premised on the nature of number inherently being an extension of “1” through a process of continual reflection of itself through all rational number. This rational number, in turn, has corresponding positive and negative values that equate to addition and subtraction as inherent entities. As inherent entities, addition and subtraction further reflect each other to form multiplication/division and exponentiation/root as inherent extensions of 1 self-reflecting. In these respects, all arithmetic is founded through a circulating mirror effect, along with number as a corresponding inseparable nature. Unlike standard arithmetic where the problem equates to one value, such as in 2 + 2 = 4, M.D.R arithmetic manifests multiple simultaneous values, usually in sets of 3 or 6 values at minimum, such as X ≡ Y ≅ (1,X,Y,A,B,C)

The structure of the mathematical symbols must be first observed in order to gain a fuller understanding. The symbol "≡" translates into English as "is congruent to" which also translates as "mirror image" from the perspective of geometry https://en.wikipedia.org/wiki/Congruence_(geometry). While "≡" is used as a symbol for modular arithmetic no rules exists within mathematics claiming a symbol can be limited to a specific interpretation. In response to the perceived mathametical/geometric problems a process of synthesis has been involved resulting in the use of "≡" as a symbol for reflection as “mirroring/mirror effect” with “≅” equating to “congruent in structure” that corresponds in value to “reflective/mirroring equality”.

1) "≡" translates as "is congruent to" in respect to modular arithmetic. "Congruent" means "in agreement or 'harmony' under a general English definition. However, it does not "strictly" translate in such a manner in the field of mathematics. The problem occurs as math and standard English, while having similarities, are not “entirely” symmetrical.

2) The geometric meaning of "congruence" further follows the same problem as stated in point 1. The problem occurs that the nature of "congruence", using "≡" has multiple different meanings at the same time in different respects (assuming we equate it to "congruence"). This works if we decide to separate mathematics, geometry, and language into separate fields however it does not solve the problem if we are trying to find a common ground in which these different fields must be "rooted" in.

3) Considering that mathematics, geometry, and language are separate and how "congruence" is portrayed in one field may differ from the other another problem occurs as they are all linked by their very nature to "axioms" as their foundation. Math, geometry, and language differ in varying degrees however, in a great twist of irony, their foundations do not as all are formed from "axioms".

4) The nature of subjective and objective realities are inherently united through "self-evidence". This self evidence breaks across all these fields while providing a common bond. So as to the common bond? I can argue a regressive or progressive argument in an attempt to unify them, however considering that each of these fields theoretically could expand ad-infinitum either a regressive or progressive argument in turn would follow that same form and function.

5) So the next question come to mind: What to do exactly? This is considering we develop these fields in such a manner where they continual differ from each other to such an extent that any reasonable form of synthetic process would either be difficult or completely impossible. Well going back to point three we have found that all still are united through the "axiom".

6) So what is the most axiomatic structure within math? What can math be reduced to in a atomic state? What can math magnify itself into as a whole? Number specifically, with all "number" being an extension of or being composed of "1".

7) And geometry, assuming the same questions? Western thinking inclines towards the line, however the problem occurs as the line exists if and only if there are "points". So the nature of the "point" being a universal form within geometry follows.

Now if one were to look at the foundation of mathematics as "number through 1" and geometry as "form through point", the next form of synthesis between Quantity (number) and "Quality" (point) would be "1" as "Point".

9)However the problem occurs as the point is viewed as a zero dimensional object, implying the point has no direction and therefore is not a thing in itself. However if the point "directs" itself into itself it becomes both 1 dimensional and simultaneously becomes a stable entity as “movement” ceases by folding upon itself.

1 exists if and only if it is "unified" or "stable". Considering both geometry and mathematics question whether their fields are "abstract" or "physical" entities, a solution can be implied as "Number as Spatial Point(s)". This understanding of space as provides the foundation for a unified physical and abstract definition of number as point(s).

10) The question occurs going back to the nature of "congruence"; How can number and space find common "ground", formative and functionally speaking? Considering all structure is founded in harmony or balance the nature of the meaning of congruence can be argued as "mirroring" or "reflecting" where the quantitative and qualitative aspects of the "Number as Spatial Point(s)" equates to an existence through a "mirroring" or "reflective" process. This nature of 1, point, and mirroring are all axiomatic processes in themselves in the respect that they are the base quantitative, qualitative and form/functional aspects of all observable physical/abstract structures.

Take for example the equation that provides the foundations of “1” as point: (1 ≡ 1) ≅ (-2,1,2):

One reflecting upon itself is congruent in structure to -2,1,2.

(1 ≡ 1) ≅ (-2,1,2)

(-2,1,2)= “angle”

One reflecting itself intradimensionally maintains itself as both stable (never changing) and unified. This act of Self-Reflection or Mirroring is equivalent to One as Point directing itself into itself. All number inherent within this first degree function manifests as part of the values produced. The point is observed as both unified and stable in 1 intradimensional nature.

This is the first degree function:

(1 ≡ 1) → 1

Simultaneously this act of intradimensional reflection manifests "2" because 1 reflecting 1 is structurally congruent to 2. 2 exists perpetually as 1 reflecting upon 1 and in this respect, exists at the same time in a different respect to 1. As one is always mirroring itself, 2 is ever present as a structural extension of 1.

This is the second degree function:

(1 ≡ 1) → 2

One mirroring itself takes a dual role of reflecting itself as both 1 and 2. In these respects 1 and 2 manifest as approximates of each other through (1 ≡ 1). As approximate points they form -1D lines to connect them. 1 point would form two corresponding negative dimensional lines to 2 as -2 or

(x ≈ y ) → -z

In these respects approximation as individuation through the -1D line which can be observed as a multiplication function resulting in negative numbers when confined to positive values.

This is the third degree function:

(1 ≈ 2 ) → -2

****We can observe the first geometric form reflected as the “angle”. Intuitively and qualitatively the angle can both be observed as the first form of measurement and the origin of measurement as a triadic structure. (expand upon and insert research)

As all number is composed upon a self-reflecting one, all 1n follows the same form and function. These sets of points and lines mirror as structural extensions of 1, through 1:

(-2,1,2) ≡ (-2,1,2) ≅ (-8,-6,-4,-3,-2,-1,0,1,2,3,4,6,8)

****with = (-2,1,2) = angle,(-3,3) = triangle,(-4,4)= square,(6,-6) = hexagon, (-8,8)= octagon

(-2,0) = angle,(-3,0)= triangle,(-4,0)= square,(-6,0) = hexagon),(-8,8) = octagon

**** All repeated numbers are valued once. For example, if 2 appears multiple times through the calculations it is observed only as “2” one time.

(1 ≡ 1) ≅ (-2,1,2)

a) (1 ≡ 1) → 1

b) (1 ≡ 1) → 2

c) (1 ≈ 2) → -2

(1 ≡ 2) ≅ (-6,-3,-2,1,2,3)

d) (1 ≡ 2 ) → (1,2)

e) (1 ≡ 2) → 3

f) (1 ≈ 2)→ -2 (1 ≈ 3)→ -3 (2 ≈ 3)→ -6

(1 ≡ -2) ≅ (-3.-2,-1,1,2,3)

g) (1 ≡ -2) → (1,-2)

h) ( 1 ≡ -2) → -1

i) (1 ≈ -1)→(-1,1) (1 ≈ -2)→(-2,2) (-1 ≈ -2)→(-2,3)

In regards to (1 ≈ -1, 1 ≈ -2); All positive numbers, as points, approximate to negative numbers, as lines, manifest -1D lines between the positive point and the negative line. In these respects all positive values, as points, multiply the number of negative values, as lines, to get a corresponding value equivalent to the lines which approximate between the point and line. “X” approximate to negative “y” always results in negative “z”.

(x ≈ -y)→(-z)

A simultaneous 1D point connects the line(s) with the line(s). These points reflect as a positive version of negative y. So if y equal -1 or -2 the corresponding value is 1 or 2.

(x ≈ -y)→(-z,y)

In regard to (-1 ≈ -2); All negative numbers, as lines, approximate to negative numbers, as lines manifest -1D line between the negative line and negative line. In these respects all negative values, as lines, multiply the number of negative values as lines, to get a corresponding value equivalent to the lines which approximate between the lines. Negative “x” approximate to negative “y” always results in negative “z”.

(-x ≈ -y)→(-z)

A simultaneous set of positive points connects the approximate lines with the corresponding line. These points are equivalent to a positive value of x and y. So if “-x” and “-y” correspond to “-1” and “-2” or if x and y are 1 and 2, the corresponding value is 3.

(-x ≈ -y)→(-z,x+y)

(2≡ 2) ≅ (-8.-4,-2,1,2,4)

h) (2 ≡ 2) → (1,2)

i) ( 2 ≡ 2) → 4

j) (1 ≈ 2)→ -2 (1 ≈ 4) → -4 (2 ≈ 4)→ -8

(2 ≡ -2) ≅ (-4,-2,0,1,2)

k) (2 ≡ -2) → (1,2,-2)

l) ( 2 ≡ -2 ) → 0

m) (1 ≈ 0)→ -1 (1 ≈ 2)→ -2 (1 ≈ -2)→ -2,2 (2 ≈ 0)→ -2 (2 ≈ -2)→ -4,2 (-2≈ 0)→2

The reflection of proportion points and lines as x ≡ -x , results in 0 as absence of reflection. In these respects what we understand of zero is that is can only be observe if and only if there is 1, however 1 can be observed without 0.

x ≡ -x → 0

Any positive (point) or negative (line) value which reflects zero, as an absence of reflection in turn observes its own existence. In these respects x stays as x and -x stays as -x.

x ≡ 0 → x

-x ≡ 0 → -x

In regards to (1 ≈ 0, 2 ≈ 0); All positive points approximate to a 0d point manifesting a -1D line that extends to zero. In these respects a negative number, as lines, is observed as proportional in value to the point. So if for example if 2 is approximate to 0, -2 as lines extends between 2 and 0. “X” approximate to zero results in negative “x”.

x≈ 0→-x

In regards to (-2≈ 0); All negative lines approximate to a 0d point manifest a 1 dimensional point the negative line(s) extend from. In these respects a positive number, as point(s), is observed as proportional in value to the line. So if for example -2 is approximate to 0, 2 as points is observed as the foundation from which -2 as line extends from. “-X” approximate to zero results in positive “x”.

-x≈ 0→x

(2 ≡ -2) ≅ (-8,-4,-2,1,2,4,6)

g) (-2 ≡ -2) → 1,-2

h) (-2 ≡ -2) → -4

i) (1≈ -2)→ -2,2 (1≈ -4)→-4,4 (-2≈ -4)→ -8,6

## Multi-Dimensional Reflective Arithmetic

### Multi-Dimensional Reflective Arithmetic

Last edited by Eodnhoj7 on Sat Dec 23, 2017 9:18 pm, edited 1 time in total.

### Re: Multi-Dimensional Reflective Arithmetic

The question occurs which came first, the form (1,2,3,etc.) or the function (+,-, etc.)?

Through this act of self-reflection, founded on both +1 and -1 respectively as intradimensional point and imaginary line, the fundamental natures of arithmetic inherently result from one reflecting into itself for arithmetic is measurement as 1 reflects itself into itself.

We can observe:

One reflecting upon itself maintains itself as an act of stability and unity. This act of Self-Reflection or Mirroring is equivalent to one as directing itself into itself. In these respects, addition manifests itself as the foundation of arithmetic through a mirroring process as the positive one.

a) 1 ≡ 1 → 1,2,-1

b) +1 ≡ +1 → +1 as addition

******Considering all multiplication exists as a second-degree version of addition, multiplication of whole numbers begins truly with "2".

The approximate nature of addition and addition, through 1 reflecting upon itself, manifests subtraction as an approximate structure. This approximate nature is the deficiency that occurs through addition reflecting upon itself to both maintain itself. Subtraction can be observed as the deficiency of addition and is not a thing in itself.

a) +1 ≡ +1 → -2 ∵ 1 ≡ 1 → (+1 ≈ +2)

*****In a separate respect considering all subtraction exists as approximation through addition cycling (reflecting) itself, subtraction is strictly a deficiency in addition as approximation.

Addition cycling (reflecting) upon itself results in multiplication as the addition of addition. This is considering that addition may be viewed strictly as a structural extension of 1 as a Positive reflecting upon itself as a Positive. This positive nature to one is structurally inherent and inseparable. Addition is a structural extension of 1, along with multiplication.

a) +1 ≡ +1 → +2,*2

*****In a separate respect multiplication exists as the individuation of -1D lines as approximation.

***** 2 x 3 = 6 = 2 + 2 + 2 = 6 with 2 ≅ 1

example: 2 ≡ *3 ≅ (-18,-12,-6,-3,-2,1,2,3,6)

a1) 2 ≡ *3 → (1,2,3) ∵ 2 ≡ 2 = (⦁ = 1,2)

a2) 2 ≡ *3 → 6 ∵ (1 ≡ 1) ≡ (1 ≡ 1) ≡ (1 ≡ 1) ≅ 6

a3) 2 ≡ *3 → ■((1 ≈ 2)→-2&(1 ≈ 3)→-3&(1 ≈ 6)→-6@(2 ≈ 3)→-6&(2 ≈ 6)→-12& (3 ≈ 6)→-18)

The second degree of Reflection, as (-2,1,2) ≡ (-2,1,2) ≅ (-8,-6,-4,-3,-2,-1,0,1,2,3,4,8) results in *2 reflecting upon itself (through 1) as exponentiation. In these regards, multiplication cycling (reflecting) upon itself results in exponentiation as the multiplication of multiplication while simultaneously maintaining addition and multiplication. Addition, multiplication and exponentiation is a structural extension of 1 reflecting itself through 2.

a) *2 ≡ *2 → (+2,*2,^4)

******Considering all exponentiation exists as a third-degree version of addition, exponentiation of whole numbers begins truly with "*2" reflecting itself as ^4.

*****2^3 = 8 = 2 x 2 x 2 = 8 with 2 ≅ 1

example: 2 ≡ *3 ≅ (-18,-12,-6,-3,-2,1,2,3,6)

a1) 2 ≡ ^3 → (1,2,3) ∵ 2 ≡ 2 = (⦁ = 1,2)

a2) 2 ≡ ^3 → 8 ∵ *(1 ≡ 1) ≡ *(1 ≡ 1) ≡ *(1 ≡ 1) ≅ 8

b1) with

a3) 2 ≡ *3 → ■((1 ≈ 2)→-2&(1 ≈ 3)→-3&(1 ≈ 6)→-6@(2 ≈ 3)→-6&(2 ≈ 6)→-12& (3 ≈ 6)→-18)

In a separate respect division manifests through -1 reflecting upon itself as /2 (subtraction reflecting itself, through 1, as division), and as the approximation of *2 and *4 (considering 4 manifests the exponentiation, it simultaneously maintains itself as multiplication.

Division can be observed as a deficiency in multiplication and not a thing in itself). In these respects, division is a biproduct of intradimensional reflection and approximation of multiplication.

a) -1 ≡ -1 → -1,/1,-2,/2

b) *2 ≡ *2 → *2,*4

b1) *2 ≡ *2 → (/8 ≅ -8 ≡ -8) ∵ *2 ≡ *2 → (*2 ≈ *4)

******Considering all division exists as a second-degree version of subtraction, division of whole numbers begins truly with "2". In a separate respect considering all division exists as approximation through multiplication cycling (reflecting) itself, division is strictly a deficiency in multiplication as approximation.

In a separate respect root manifests through /2 reflecting upon itself as √4 or division cycling (reflecting) upon itself as root. Subtraction, Division and Root is a structural extension of 1 reflecting itself through 2.

Through this act of self-reflection, founded on both +1 and -1 respectively as intradimensional point and imaginary line, the fundamental natures of arithmetic inherently result from one reflecting into itself for arithmetic is measurement as 1 reflects itself into itself.

We can observe:

One reflecting upon itself maintains itself as an act of stability and unity. This act of Self-Reflection or Mirroring is equivalent to one as directing itself into itself. In these respects, addition manifests itself as the foundation of arithmetic through a mirroring process as the positive one.

a) 1 ≡ 1 → 1,2,-1

b) +1 ≡ +1 → +1 as addition

******Considering all multiplication exists as a second-degree version of addition, multiplication of whole numbers begins truly with "2".

The approximate nature of addition and addition, through 1 reflecting upon itself, manifests subtraction as an approximate structure. This approximate nature is the deficiency that occurs through addition reflecting upon itself to both maintain itself. Subtraction can be observed as the deficiency of addition and is not a thing in itself.

a) +1 ≡ +1 → -2 ∵ 1 ≡ 1 → (+1 ≈ +2)

*****In a separate respect considering all subtraction exists as approximation through addition cycling (reflecting) itself, subtraction is strictly a deficiency in addition as approximation.

Addition cycling (reflecting) upon itself results in multiplication as the addition of addition. This is considering that addition may be viewed strictly as a structural extension of 1 as a Positive reflecting upon itself as a Positive. This positive nature to one is structurally inherent and inseparable. Addition is a structural extension of 1, along with multiplication.

a) +1 ≡ +1 → +2,*2

*****In a separate respect multiplication exists as the individuation of -1D lines as approximation.

***** 2 x 3 = 6 = 2 + 2 + 2 = 6 with 2 ≅ 1

example: 2 ≡ *3 ≅ (-18,-12,-6,-3,-2,1,2,3,6)

a1) 2 ≡ *3 → (1,2,3) ∵ 2 ≡ 2 = (⦁ = 1,2)

a2) 2 ≡ *3 → 6 ∵ (1 ≡ 1) ≡ (1 ≡ 1) ≡ (1 ≡ 1) ≅ 6

a3) 2 ≡ *3 → ■((1 ≈ 2)→-2&(1 ≈ 3)→-3&(1 ≈ 6)→-6@(2 ≈ 3)→-6&(2 ≈ 6)→-12& (3 ≈ 6)→-18)

The second degree of Reflection, as (-2,1,2) ≡ (-2,1,2) ≅ (-8,-6,-4,-3,-2,-1,0,1,2,3,4,8) results in *2 reflecting upon itself (through 1) as exponentiation. In these regards, multiplication cycling (reflecting) upon itself results in exponentiation as the multiplication of multiplication while simultaneously maintaining addition and multiplication. Addition, multiplication and exponentiation is a structural extension of 1 reflecting itself through 2.

a) *2 ≡ *2 → (+2,*2,^4)

******Considering all exponentiation exists as a third-degree version of addition, exponentiation of whole numbers begins truly with "*2" reflecting itself as ^4.

*****2^3 = 8 = 2 x 2 x 2 = 8 with 2 ≅ 1

example: 2 ≡ *3 ≅ (-18,-12,-6,-3,-2,1,2,3,6)

a1) 2 ≡ ^3 → (1,2,3) ∵ 2 ≡ 2 = (⦁ = 1,2)

a2) 2 ≡ ^3 → 8 ∵ *(1 ≡ 1) ≡ *(1 ≡ 1) ≡ *(1 ≡ 1) ≅ 8

b1) with

a3) 2 ≡ *3 → ■((1 ≈ 2)→-2&(1 ≈ 3)→-3&(1 ≈ 6)→-6@(2 ≈ 3)→-6&(2 ≈ 6)→-12& (3 ≈ 6)→-18)

In a separate respect division manifests through -1 reflecting upon itself as /2 (subtraction reflecting itself, through 1, as division), and as the approximation of *2 and *4 (considering 4 manifests the exponentiation, it simultaneously maintains itself as multiplication.

Division can be observed as a deficiency in multiplication and not a thing in itself). In these respects, division is a biproduct of intradimensional reflection and approximation of multiplication.

a) -1 ≡ -1 → -1,/1,-2,/2

b) *2 ≡ *2 → *2,*4

b1) *2 ≡ *2 → (/8 ≅ -8 ≡ -8) ∵ *2 ≡ *2 → (*2 ≈ *4)

******Considering all division exists as a second-degree version of subtraction, division of whole numbers begins truly with "2". In a separate respect considering all division exists as approximation through multiplication cycling (reflecting) itself, division is strictly a deficiency in multiplication as approximation.

In a separate respect root manifests through /2 reflecting upon itself as √4 or division cycling (reflecting) upon itself as root. Subtraction, Division and Root is a structural extension of 1 reflecting itself through 2.

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