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Basic Arithmetic is Recursion; Number as Recursive 0; Identity as Relation; Number as Spatial Process
There is addition x+x.
There is multiplication as the addition of addition:
the number times x is scaled by addition, times being the scale of X as subject to the number of times the number X scales on itself through addition, it is higher level addition, the existence of addition at a new repeating scale, thus the recursion of addition:
X = X×1
X+X = X×2
X+X+X = X×3
X+X+X+X = X×4
There is exponents, the multiplication of the number of times a number multiplies itself, the multiplication of multiplication. Higher level multiplication with multiplication being a higher level of addition…thus addition exists across scales repeatedly
Each case is a recursive scaling of the operation of addition.
Subtraction, division, roots are of course the inverse.
Subtraction is X - Y
There is division as the subtraction of subtraction, higher level subtraction:
the number times x is scaled by subtraction, times being the scale of X as subject to the number of times the number X scales on itself through subtraction, it is higher level subtraction, the existence of subtraction at a new repeating scale, thus the recursion of subtraction:
X = X/1
X-Y-Y = X/2
X-Y-Y-Y = X/3
X-Y-Y-Y-Y = X/4
There is roots, the division of the number of times a number divides itself, the division of division by scaling of division. Higher level division with division being a higher level of subtraction…thus subtraction exists across scales repeatedly
Each case is a recursive scaling of the operation of addition or subtraction.
In these respects:
There is addition, the addition of addition is multiplication, the multiplication of multiplication is exponents; there is substraction, the subtraction of subtraction is division, the division of division is roots. Thus what occurs is the scaling of an operation of addition and subtraction thus a recursion of said operation at scale.
The collapse of the operator and operand dichotomy reveals a change in how the number becomes distinct and is revealed as a patterned process where form is function and function is form. This will be explained later where numbers are viewed as spatial processes.
A postive number is addition by nature, a negative number is subtraction; operator and operand are reduced to the single distinction of a postive and/or negative number.
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+X+Y = X+Y
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+X-Y = X-Y
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-X+Y = Y-X
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-X-Y = -X±Y
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+X++Y = X×Y
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+X–Y = X/Y
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–X+Y = Y/X
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-X—Y = -X/-Y
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—X-Y = -Y/-X
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+X++++Y = X^Y
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+X----Y = YrtX
The identity of number thus is the distinction of it by the limits of itself. In these respects number can be reduced purely to spatial events by degree of a non-axiomatic state where distinction is the emergent pattern.
0 can be observed simultaneously as a 0d point for this exercise: ●
0 is nothing on its own terms; ●
The recursion of ● results in the distinction of ●, by contrast to itself, while simultaneously being self-contained as its own limit: ●●
The distinction of ●● is 1, this can be seen in a line segment.
●●● is 2, with a corresponding -1 that results by degree of the absence of unity, the space of difference as negative space between 1 and 2.
●●●● is 3 with a corresponding -2.
So on and so forth.
Thus with the recursion of 0 as ● comes the corresponding natural numbers.
Addition is compounding of line segments; subtraction is ghs compounding of a postive and negative line segment.
Fractions are effectively the contraction of one set of line segments to another, multiplication is the inverse expansion of one set of line segments through another.
What remains is linear folding at the primitive level. Corresponding symmetrical n-d forms observes this at a more advanced level.
In these respects each line segment is a relative variable as it contains further line segments, each 0 sequence is a variable as it contains further 0 sequences; in these respects all finite numbers are finite infinities; all irrational numbers are effectively a process of non-terminating recursion relative to a terminated recursion.
This recursivd nature results in the collapse of numbers into sets when observed from another angle.
What remains is number as process of recursive 0. Basic arithmetic is the relation of recursive 0 sequences (line segments or n-d forms). In these respects identity is reduced to relation prior to equality thus what remains of identity is pure pattern as distinction.
All numbers are thus inseperable from patterns themselves, by degree of all degrees of recursion containing the symmetry of ●● at all levels of the sequence and correspondingly resulting in ●● as a fixed point thus results in further scaling of fixed points (●●●, ●●●●, etc.)