## Multi-Dimensional Reflective Arithmetic

### Multi-Dimensional Reflective Arithmetic

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

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

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.

### Re: Multi-Dimensional Reflective Arithmetic

Can you summarise your argument in 500 words or less?

### Re: Multi-Dimensional Reflective Arithmetic

All number exists as reflective space, with 1 being an intradimensional point that exist ad-infinitum through a mirroring process that constitutes space itself as space, while simultaneously providing the foundations for "space as dimension through direction". Considering the point exists as unified and everchanging, we observe it locally in space/time through approximation in which the points (as extension of the point) exist through the connection of -1 dimensional lines that are: imaginary, negative in dimension (deficient in dimension but existing the the 1d point) and provide the foundation for what we understand of as deficiency in structure, aka randomness, as the limit of unity. This points mirrors itself ad-finitum at such a high rate, that it does not move while simultneously relflecting the dualistic understanding of infinity as spatial "limit" and "no-limit" with the third dimension being...well dimension itself as "direction".

Because the 1d point is the foundation for all reality, including consciousness (in which we intuitively "number" the 0d point contradictory) it is the foundation for what we understand of as number. 1 as positive cannot be seperated from the point, as it is the point. In its positive nature, as equal to addition as summation, if reflects upon itself to maintain +1. Simultaneously, addition as + which is inherent and inseperable from the number, reflects to form multiplication as the addition of addition or *1. +1 also reflects to form +2 and *2.

So where standard addition observes 1+1=2, reflective arithmetic observes both this and an inherent "set" with which the equation is composed:

+1 ≡ +1 ≅ {+1,*1,+2,*2}

****This post is old and the symbols used have been changed to avoid confusion. I have to update the post when I am finished with the calculations, several of the equations in the above text are void.

This set in turn existing as points, or lines if negative numbers, approximates. This approximation function observes that each point must find the corresponding connection between them. This approximation as connection, in turn results as the negative number (equivalent to negative dimensions as the line), wich both connects the points and exists as subtraction and division. Subtraction merely being the approximation of addition, division the approximation of multiplication, and subtraction mirroring subtraction to form division.

All positive arithmetic functions are inseperable from the point, as 1d space (not 0d space). All negative arithmetic functions are inseperable from the line as -1d space.

I will cut this out for brevity, assuming questions for the first part, however the equation about converting geometric solids gives a glimpse of the calculations. I will cover approximation later, as the negative dual to the reflective portion, if you wish.

This does not argue against the standard foundations of mathematics and geometry (founded in the 0d point and 1d line) but observes them as foundations for "relativism" or "relation" in which the symbols exist if and only if they "relate". In these respects standard mathematics is founded, and at its peak, from relativistism as the relation of parts that exist as 1d linear spaces individuating through 0d point.

In simple terms, from the perspective of an ethereal binding space, all numbers exist as positive points and negative lines which are inseperable from sets rooted in 1 as 1, while providing a foundation for both number and arithmetic as an inherent mirroring space that acts as binding median through the promulgation of symmetry.

Sorry if over the 500 words, but their is alot the cover. The paper alone is 30+ pages.

### Re: Multi-Dimensional Reflective Arithmetic

Yes. Very interesting topic. Not sure how it relates to your earlier posts in this thread. To be fair I didn't read them. But still. Your latest post has the virtues of clarity and brevity.

The subject of nonconstructive proof is so interesting, since many proofs in higher math are nonconstructive. That is, we can show that some object exists, but we have no way to explicitly construct an example of one.

### Re: Multi-Dimensional Reflective Arithmetic

If form follows functions, we may be able to construct an object by constructing a function.wtf wrote: ↑Mon Jun 11, 2018 11:32 pmYes. Very interesting topic. Not sure how it relates to your earlier posts in this thread. To be fair I didn't read them. But still. Your latest post has the virtues of clarity and brevity.

The subject of nonconstructive proof is so interesting, since many proofs in higher math are nonconstructive. That is, we can show that some object exists, but we have no way to explicitly construct an example of one.

### Re: Multi-Dimensional Reflective Arithmetic

Few issues need worked out, but here is a general update on Mirror Theory with some basic examples:

Axioms:

A) All numbers are inseparable from there arithmetic functions, with these arithmetic functions inherent within the number itself.

B) All mirroring numbers maintain themselves as part of the inherent set. These mirroring numbers as extensions of 1 by default maintain 1 as an inherent element of the set.

b1)This can be seen in Step A.

C) All arithmetic functions inherent within the numbers composed both the number an eachother.

c1) All additive functions, marked by (∙), mirror into a multiplicative function, marked by (:). Multiplication is the addition of addition, with multiplication having an inherent element of addition in it.

c2) All multiplicative functions, marked by (:), mirror into a power function, marked by (⁞). Powers are the multiplication of multiplication, with powers having an inherent element of addition in it through multiplication.

c3) All mirrored functions have an inherent element of 1 corresponding to that function which is mirrored into the numbers.

c4) This can be seen in Step B.

D) All numbers mirror each other in accordance with their inherent function. If the function is a mirror of base addition or subtraction, such as multiple or powers, this mirroring process contains as an element the basic functions which compose it.

d1) Considering the arithmetic function is inherent within the answer, where the arithmetic will be the same, the corresponding number as the answer will mirror in structure the arithmetic function in which it is composed.

d2) This corresponding answer will contain as an element 1 and its corresponding arithmetic properties

d3) This can be seen in Step C and D along with their corresponding sub-steps

E) All positive numbers are called point numbers, in reference to positive 1 (as additive), being the point of origin. The resulting numbers from steps A, B, C/D and their corresponding substeps resulted in a set. All repeated numbers are observed once. In these respects mirror theory observes the mirroring of numbers resulting in sets of numbers whose inherent elements provide the foundation for arithmetic functions. All numerical form and function is premised as an extension of 1.

Example Set 1:

⨀ (∙1,∙1) ⧂ (∙1, :1, ∙2, :2)

a) (∙1, ∙1) → ∙1

b) (∙1, ∙1) → :1 ∋ ∙1

c) (∙1, ∙1) → (:2 ∋ (:1 ∋ ∙1)) ∋ (∙2 ∋ ∙1)

⨀ (∙2,∙3) ⧂ (∙1, :1, ∙2, :2, ∙3, :3, ∙5, :5)

a) (∙2, ∙3) → ∙2 ∋ ∙1 , ∙3 ∋ ∙1

b) (∙2, ∙3) → :2 ∋ (:1 ∋ ∙1), :3 ∋ (:1 ∋ ∙1)

c) (∙2, ∙3) → (:5 ∋ (:1 ∋ ∙1)) ∋ (∙5 ∋ ∙1)

⨀ (∙2,:3) ⧂ (∙1, :1, ∙2, :2, ∙3, :3, ∙5, :5, ∙6, :6)

a) (∙2, :3) → ∙2 ∋ ∙1, (:3 ∋ (:1 ∋ ∙1)) ∋ (∙3 ∋ ∙1)

b) (∙2, :3) → :2 ∋ (:1 ∋ ∙1), :3 ∋ (:1 ∋ ∙1)

c) (∙2, :3 ∋ ∙3) → (:5 ∋ (:1 ∋ ∙1)) ∋ (∙5 ∋ ∙1)

c) (∙2, :3) → (:6 ∋ (:1 ∋ (:1 ∋ ∙1)) ∋ (∙6 ∋ ∙1)

⨀ (:2,:3) ⧂ (∙1, :1, ⁞1, ∙2, :2, ⁞2, ∙3, :3, ⁞3, ∙5, :5, ∙6, :6, ⁞6)

a) (:2, :3) → (:2 ∋ (:1 ∋ ∙1)) ∋ (∙2 ∋ ∙1), (:3 ∋ (:1 ∋ :1)) ∋ (∙3 ∋ ∙1)

b) (:2, :3) → ((⁞2 ∋ (⁞1 ∋ (:1 ∋ ∙1)) ∋ (:2 ∋ (:1 ∋ ∙1))) ∋ (∙2 ∋ ∙1)

b1) ((⁞3 ∋ (⁞1 ∋ (:1 ∋ ∙1)) ∋ (:3 ∋ (:1 ∋ ∙1))) ∋ (∙3 ∋ ∙1)

c) (:2, :3) → ((⁞6 ∋ (⁞1 ∋ (:1 ∋ ∙1)) ∋ (:6 ∋ (:1 ∋ ∙1))) ∋ (∙6 ∋ ∙1)

C1) (:2 ∋ ∙2, :3 ∋ ∙3) → (:5 ∋ (:1 ∋ ∙1)) ∋ (∙5 ∋ ∙1)

Axioms:

A) All numbers are inseparable from there arithmetic functions, with these arithmetic functions inherent within the number itself.

B) All mirroring numbers maintain themselves as part of the inherent set. These mirroring numbers as extensions of 1 by default maintain 1 as an inherent element of the set.

b1)This can be seen in Step A.

C) All arithmetic functions inherent within the numbers composed both the number an eachother.

c1) All additive functions, marked by (∙), mirror into a multiplicative function, marked by (:). Multiplication is the addition of addition, with multiplication having an inherent element of addition in it.

c2) All multiplicative functions, marked by (:), mirror into a power function, marked by (⁞). Powers are the multiplication of multiplication, with powers having an inherent element of addition in it through multiplication.

c3) All mirrored functions have an inherent element of 1 corresponding to that function which is mirrored into the numbers.

c4) This can be seen in Step B.

D) All numbers mirror each other in accordance with their inherent function. If the function is a mirror of base addition or subtraction, such as multiple or powers, this mirroring process contains as an element the basic functions which compose it.

d1) Considering the arithmetic function is inherent within the answer, where the arithmetic will be the same, the corresponding number as the answer will mirror in structure the arithmetic function in which it is composed.

d2) This corresponding answer will contain as an element 1 and its corresponding arithmetic properties

d3) This can be seen in Step C and D along with their corresponding sub-steps

E) All positive numbers are called point numbers, in reference to positive 1 (as additive), being the point of origin. The resulting numbers from steps A, B, C/D and their corresponding substeps resulted in a set. All repeated numbers are observed once. In these respects mirror theory observes the mirroring of numbers resulting in sets of numbers whose inherent elements provide the foundation for arithmetic functions. All numerical form and function is premised as an extension of 1.

Example Set 1:

⨀ (∙1,∙1) ⧂ (∙1, :1, ∙2, :2)

a) (∙1, ∙1) → ∙1

b) (∙1, ∙1) → :1 ∋ ∙1

c) (∙1, ∙1) → (:2 ∋ (:1 ∋ ∙1)) ∋ (∙2 ∋ ∙1)

⨀ (∙2,∙3) ⧂ (∙1, :1, ∙2, :2, ∙3, :3, ∙5, :5)

a) (∙2, ∙3) → ∙2 ∋ ∙1 , ∙3 ∋ ∙1

b) (∙2, ∙3) → :2 ∋ (:1 ∋ ∙1), :3 ∋ (:1 ∋ ∙1)

c) (∙2, ∙3) → (:5 ∋ (:1 ∋ ∙1)) ∋ (∙5 ∋ ∙1)

⨀ (∙2,:3) ⧂ (∙1, :1, ∙2, :2, ∙3, :3, ∙5, :5, ∙6, :6)

a) (∙2, :3) → ∙2 ∋ ∙1, (:3 ∋ (:1 ∋ ∙1)) ∋ (∙3 ∋ ∙1)

b) (∙2, :3) → :2 ∋ (:1 ∋ ∙1), :3 ∋ (:1 ∋ ∙1)

c) (∙2, :3 ∋ ∙3) → (:5 ∋ (:1 ∋ ∙1)) ∋ (∙5 ∋ ∙1)

c) (∙2, :3) → (:6 ∋ (:1 ∋ (:1 ∋ ∙1)) ∋ (∙6 ∋ ∙1)

⨀ (:2,:3) ⧂ (∙1, :1, ⁞1, ∙2, :2, ⁞2, ∙3, :3, ⁞3, ∙5, :5, ∙6, :6, ⁞6)

a) (:2, :3) → (:2 ∋ (:1 ∋ ∙1)) ∋ (∙2 ∋ ∙1), (:3 ∋ (:1 ∋ :1)) ∋ (∙3 ∋ ∙1)

b) (:2, :3) → ((⁞2 ∋ (⁞1 ∋ (:1 ∋ ∙1)) ∋ (:2 ∋ (:1 ∋ ∙1))) ∋ (∙2 ∋ ∙1)

b1) ((⁞3 ∋ (⁞1 ∋ (:1 ∋ ∙1)) ∋ (:3 ∋ (:1 ∋ ∙1))) ∋ (∙3 ∋ ∙1)

c) (:2, :3) → ((⁞6 ∋ (⁞1 ∋ (:1 ∋ ∙1)) ∋ (:6 ∋ (:1 ∋ ∙1))) ∋ (∙6 ∋ ∙1)

C1) (:2 ∋ ∙2, :3 ∋ ∙3) → (:5 ∋ (:1 ∋ ∙1)) ∋ (∙5 ∋ ∙1)

### Re: Multi-Dimensional Reflective Arithmetic

Sometimes we show a given function exists but we can't construct it. A well-ordering of the reals, for example. Any proof that depends on the axiom of choice (of which the well-ordering of the reals is an example).

### Re: Multi-Dimensional Reflective Arithmetic

I don't believe that to be the case if we use a set of functions to compose other functions...now will what I say be said in a math book? Probably not.

### Re: Multi-Dimensional Reflective Arithmetic

If you have an explicit well-ordering of the real numbers you'll become famous.Eodnhoj7 wrote: ↑Tue Jun 12, 2018 7:18 pmI don't believe that to be the case if we use a set of functions to compose other functions...now will what I say be said in a math book? Probably not.

The axiom of choice itself posits the existence of a set that can't possibly be constructed in any meaningful way. Are you denying that nonconstructive objects exist?

Are you therefore arguing a constructivist position? Or are you just unaware of the use of nonconstructive proof in standard math?

### Re: Multi-Dimensional Reflective Arithmetic

wtf wrote: ↑Tue Jun 12, 2018 8:03 pmIf you have an explicit well-ordering of the real numbers you'll become famous.

The axiom of choice itself posits the existence of a set that can't possibly be constructed in any meaningful way. Are you denying that nonconstructive objects exist?

The axiom of choice is dependent upon, pardon the pun, a "choice" in the means of measurement, which in turn (and you can correct me if I am misinterpreting any of the above, would cause the problem of meaning in regards to forming the correct function considering the function can begin just about anywhere and in turn lead to an infinite number functions. The problem occurs in the axiom of choice, is that while the point of measurement can begin anywhere relativistically it is dependent upon a non-constructivist object of "1" in respect that the choice is at minimum quantifiable as "1".

A paradox ensues between the non-constructivist and constructivist position, and jump on in if you believe I am missinterpretting something, where the non-constructivist position is at minimum dependent upon the object of "1" and the constructivist position (as the means to form objects through measurement as a function) is dependent upon "choice" as 1. In these respects 1 is a neutral form and function that provides the center point of origin in regards to the constructivist and non-constructivist position and in these regards must be argued (and I am working on this as we speak) as both form and function.

Mirror theory observes this 1 as both form and function, however I am getting the impression I will need to draft a set of equations as axioms if my point is to be given further credit. The problem occurs with the non-constructivist and constructivist dualism that the functions in themselves, as evidenced in base arithmetic, are composed of eachother and take on a dual role as quantitative object (mirror theory observed this):

The addition of addition, results in multiplication.

The multiplication of multiplication, as the addition of addition of addition of addition, results in powers.

Dually,

The substraction of subtraction results in division.

The division of division, as the subtraction of substraction of subtraction of subtraction, results in roots.

In these respects:

+1 → (*1=(+1+1)=+2) → (^1= (*1*1) = (+1+1+1+1) =+4)

-1 → (/1=(-1-1)=-2) → (1r= (/1/1) = (-1-1-1-1) =-4)

In these respects 1 as positive, can be argued as 1 as addition. 1 as addition, when mirrored (directed through itself as repitition resulting in further form and function by allowing a multiplicity in form in function) results in 1 as a multiple and 1 as a power. The same logic applies for negative 1 as subtractive 1.

The question of form and function in quantity is then implied as fundamentally a repitition through direction which we see as a standard inherent function within the number line itself. 1 exists through its ability direct itself, ad-infinitum, to infinity, with 1 existing as direction itself through the inherent arithmetic function where postive and negative correspond as the corresponding 7th and 8th elements to the arithmetic functions as:

1) Positive: Addition/Multiplication/Powers

2) Negative: Substraction/Division/Roots

Are you therefore arguing a constructivist position? Or are you just unaware of the use of nonconstructive proof in standard math?

Upon what I am aware of, I lean towards more of a constructivist position...but the split is almost even...the argument above should give some definition as to why.

### Re: Multi-Dimensional Reflective Arithmetic

"This is in contrast to a non-constructive proof (also known as an existence proof or pure existence theorem) which proves the existence of a particular kind of object without providing an example."

The basic proof for an object stems from the law of identity as 1=1 or "∃1" where 1 exists as itself...is there something I am missing considering it depends on the existence theorem?

### Re: Multi-Dimensional Reflective Arithmetic

In classical set theory, a mathematical object exists when the axioms, or a theorem derived from the axioms, say it does. So for example, how do we know there's an infinite set? The Axiom of Infinity says so. Take away the axiom of infinity and you have no infinite sets.

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