## The Paradox of Point Division and a Theoretical 1d Point?

Philosophical, mathematical and computational logic, linguistics, formal argument, game theory, fallacies, paradoxes, puzzles and other related issues.  ### The Paradox of Point Division and a Theoretical 1d Point?

The Paradox of Point Division x/y → (1+y)*x = z

Presented argument:

What we understand of the point as a 0d entity causes problems when we refer to it as a unit of measurement. For example if a line is divided by three points we intuitively quantify 0 into a unit conducive to "1".

The problem occurs in the act of quantifying 0 we inherently view it as a unit in itself, however a unit (as a part or piece of something greater, usually another unit) relies on some inherent degree of dimensionsality. This dimensionality, often defined as the minimum number of coordinates or something conducive to a spatial property in itself, is dependent upon viewing space in directional terms (premised often times in the 1d line).

The problem occurs in the respect we can only view the extradimensional line in terms of relation, it must relate to other lines in order to exist. This can be observed through angles.

Standard division observes the individuation of a unit into parts, with the original unit being the potential relation of those parts with the parts themselves being the actual relations as ω/φ.

In these respects what we understand of division is actually an observation of particulation as change through actual and potential relations.

The question occurs as to what happens when we view the unit, in this case "1", as a point in itself? How can a point be divided into anything less than a point?

Now the point as "1" appears to form somewhat of a contradiction considering we view it, in standard academic terms, as a 0 dimensional entity. The problem of point division seems, in these respects, to end in a contradiction as 0/x always results in 0. Viewing the point as 1 would require it to contain dimensionality with this dimensionality fundamentally being intradimensional in nature. In simple terms, the point folding into itself ad-infinitum to form the foundation of dimension as "direction" through an intradimensional nature.

So where the 1d line as a spatial "unit" is dependent upon an infinite extradimensionality, the 1d point would be dependent upon infinite intradimensionality as spatial "unity".

Observing the point in these terms, as intradimensional with the 0d point being a dual "non-being", what we observe as division fundamentally changes reflective of the premises.

A point divided through 1 line of negative dimension results in the point as “2”.

This can be observed from a simple mental exercise where one imagines a point with a line halving it. The point is "halved", however considering the point is still a point, this results in the 1 point turning into two points. A positive 1d point, with positive equated to "summation as unity" similar to how we view addition as a summative process resulting in unity, mirroring a divisive -1d dimensional line, with negative dimensionality being synonymous to both an imaginary unit and one of "deficiency" or "absence", results in the divisive -1d line halving the point. The -1d line is not a dimension in itself but rather an approximation between the point as point, considering the point can only be observed approximately and not as a whole.

Considering a point cannot be halved, as the division of a point still results in the point, what we observe is a form of multiplication.

Take for another example the biological cell. When it splits to reproduce, does it divide itself or multiply itself? Or both? Either way we see a process of individuation.

Furthermore, a point divided through the mirroring of divisive 2, in which -1 dimensional line “halves” the point 2 times, results in the point as three. Where the standard act of division results in a fraction of the original, what we observe here is an approximation of 1 point through multiplicity. This approximation observes these parts in themselves as extension of a point as points.

In these respects the divisive value merely acts as a set of-1d lines connecting multiple points.

It is in this understanding of point division that the algebraic expression of standard division “x/y” is equivalent to “(1+y)*x = z” in which “y” equals the number of divisive -1d lines, “x” equals the number of sets connected by these lines and “z” equals the total number of points.

In these respects a point divided is actually a point multiplied as the point itself remains as a point. Multiplication and division are identical duals relative to the point as "1".

Agree/Disagree Why?
Eodnhoj7
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Posts: 102
Joined: 02 Mar 2018    ### Re: The Paradox of Point Division and a Theoretical 1d Point

What if the 0d point is inverted into an intradimensional entity as the 1d point? The problem occurs that in the foundations of Euclidian and Non-Euclidian geometry no 1d point occurs, it is not even a concept. As a matter of fact, according to modern instinct, it appears neither as a contradiction or paradox as it is not even a concept. But is it really a concept we have not fully dealt with or categorized? Intuitively the 0d point is quantified into units of 1, in various phases of measurement, yet the multiplication or division of zero, with this individuation being the foundations of measurement, cannot occur as 0 can neither be multiplied or divided without resulting in zero. To observe 0 as 1 unit is to equate it to a unit which can be multiplied or divided, and yet current mathematics does not allow for this function. Contradictory it is often times quantified and qualified in such terms.

However if we look at the point as intradimensional, or directed into itself, what we get is a new entity that acts as unified totality that exists on its own terms and is purely axiomatic both qualitatively and quantitatively. In one respect the point as intradimensional is directed into itself, with the point being a center that allows for infinity, or absence of limit. The point as both center to other centers and center to its own center, maintains an spatial dimensionality of “no-limit” as a center contains no limit. We can observe this within transcendental qualities and quantities such as Pi.(quote) In these respects the center exists as its own axiom, through the point, and holds the definition of space as “no-limit”.

However, the problem occurs in regards to this definition of “no-limit”: Can the point mediate between “no-limit” and “limit”? The solution appears to exist within the concept of intradimensionality. The 1d point, as intradimensional, maintains itself as a “limit” through dimensionality with dimension merely being space as direction. In simpler terms dimensionality is merely space as direction. We can observe this in the previously mentioned extradimensional space, in which dimensions must progress past their origins through relations with further dimensions, i.e the 1d line relative to the 1d line through the 0d point. Dimension in these respects, appears to exists as a boundary in itself as the relation of dimensions is strictly the relations of “direction”.

From this premise the question occurs as to how the point, as intradimensional, can have direction if direction implies an extradimensional spatial relation in which something is directed towards something? In simpler terms dimension unavoidably requires relation with relation as the observation of an inherent unit-particulate separation to some degree. If the 1d point is directed into itself, where would it move? The 1d point cannot be empty otherwise it requires the 0d point to exist inside it as a separate dimension, or maybe better put “absence” of dimension. If this is the case the 1d point is no longer a point but rather a circle or sphere. The “center” nature of the 1d point justifies its nature of “no-limit” as the intradimensional nature is without limit through the “no-limit” of the center. In simpler terms, because the point is “center” it is directed into itself without limit.

This nature of center as “no-limit” simultaneously justifies the intradimensional nature of the point as “limit”. Considering dimension exists as directional space which forms limit and boundary, the infinite intradimensional nature of the 1d point, observes the unlimited direction of space as unlimited “limit”. “Limit” in turn exists on its own terms as ever-present through the 1d point. The 1d point is infinite on its own terms and exists as a medial unifying space qualitatively as “limitless limit” and quantitatively as “numberless number” (quote).

The 1d point directed into itself, observes space as direction existing ad-infinitum as not the movement towards a center but rather the center itself being pure unified direction which moves itself paradoxically into non-movement or stability. As the 1d point alone exists, it has nowhere to move except to itself, and in these respects the rate of movement exists ad-infinitum. To illustrate this point, take for example a spinning wheel. When the wheel spins slowly, the movement can be observed through the various senses especially sight and sound. As the movement increases the change in sensual perception occurs as the speed causes a distortion in the original appearance of the wheel, while the squeaking tones increase to a higher pitch. The increase in speed causes an increase in sound pitch and observable movement. Eventually the rate, in this case infinity, causes the wheel not only to cease what appears as movement but the pitch reaches a frequency which cannot be heard let alone identitified. Movement ad-infinitum paradoxically results in an absence of movement as infinite movement can go nowhere except to itself as infinity. Considering infinity exists through, but is not limited to all existence, the 1d points contains all phenomena as 1 eternal ever-present moment.
Eodnhoj7
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Posts: 102
Joined: 02 Mar 2018   