Structure of Space

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ne might inquire about the composition of the space between two points on a line because it is understood that between any two points there are an infinite number of other points. There are two types of points to be considered:
Those representing rational numbers and
those representing irrational numbers.

Rational numbers, specifically integers, will represent the 'countable' points while irrational numbers will represent 'uncountable' numbers (a higher infinity by Cantor's Diagonal Proof). Thus, an infinite quantity of irrational numbers will always fit between two adjacent rational numbers. Which is required for the construction of a manifold of differentiable things.

The integers are quantitive and can be expressed abstractly as a binary number e.g. 100100111010010, a series of ones and zeroes in a line having a beginning and end.

The irrational numbers are represented by an infinite string of random 1's and 0's and cannot in principle denote a quantity. They are qualitive. They denote a specific order of objects.

The finite geometric embodiment of these two types is a line for quantity and a circle for quality. Thus, if we place a finite number of 1's and 0's on the rim of a circle, there is neither beginning nor end and the string carries qualitive information about the order of the symbols only.

Ponder for a moment that a ray (quantitive) has a beginning and no end. While a line with neither beginning nor end (qualitive) is logically equivalent to a circle of infinite radius.

A ray advancing outward encompassing 'countable' points (the integer count necessitated at the beginning) must have its points differentiated by space (an entity which cannot be displayed in less than the two dimensions required by a circle).

By placing the circle (space) at right angles to the ray (integer count), the identity of each fundamental concept is preserved. Ultimately, this is the reason that there are three physical dimensions.

And, in fact, if we turn our number one sideways and penetrate the zero with it, we have an excellent abstract symbol for existence.



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