odysseus wrote: ↑Tue Mar 06, 2018 2:55 am
Eodnhoj7
What if number, as premised in space it based upon an intradimensional point where it mirrors itself ad-infinitum as both limit (direction of space through intradimensionality) and no-limit (infinity as the 1d point is pure direction?)
I would probably ask you, what do you mean by "intradimensional point"? What do you mean by a one dimensional point? And so on. Such things likely belong to science fiction rather than meaningful utterances. You would need to shoe why I should take these seriously.
The paradox of point division is one example:
http://www.sciencechatforum.com/viewtop ... 65&t=34194
And another argument as an excerpt from part of a paper I am working on...the whole argument is not presented but should give you a based idea:
The first axiom of the argument is a simple one: what we understand of all phenomena, at minimum, is strictly an observation of space. It is the one universal axiom which exists as the limits that give structural boundaries to reality. In a separate and simultaneous respect, it is absent of the very same definition as no-limit. Space observes a dualistic nature of limit and no-limit.
Observing the definition of space, within any given dictionary source, one is placed into a paradox. A whole host of definitions are given, which include but are not limited to: “area” “volume”, “dimension”, and “limit” These definitions,summated under the last definition as “limit”, reflect back upon the process of definition as a form of limit in itself by which a phenomena exists through the inherent limits which form it. Space as limit is limit through space, with the observation of any dictionary definition resulting in a dualistic circular and linear form of reasoning where one definition leads to another while simultaneously circling back to the original. Under these terms circular rational is justified through inherent linear elements and vice versa while observing, under certain degrees, Mirimanoff’s concept of “wellfoundedness” in which the definition as a set of information contains no infinite descension further implying an original source.
This dualism of progressive linear and circular definition provides a limit in itself through a process of mirroring in which the further corresponding definitions in turn follow this same process. The axiom of space follows this definition process in which a limit reflects itself through a further limit, rationally in both form and function as circular, and reflects further limits, rationally in both form and function as linear, in which an observation of no-limit occurs. This observation of “no-limit” is founded inherent within the dictionary definitions of space in an immediate respect within the aforementioned definition itself. In a separate respect, function follows form where these definitions reflect through further definition ad-infinitum in a dual circular and linear regressive/progressive manner. Limit and No-Limit are dependent on a dual form of circular and linear reasoning that simultaneously manifests further definition while maintaining there own under spatial terms.
This dualistic understanding of space can further be observed in many of the works of the pre-socratic including but not limited to the Pythagoreans and Anaximander. The Pythagorean Philolaus observed “that all things in the universe result from a combination of the unlimited and the limiting; for if all things had been unlimited, nothing could have been the object of cognizance.” Aristotle observed “[the Pythagoreans] plainly say that when the one had been constructed, whether out of planes or of surface or of seed or of elements which they cannot express, immediately the nearest part of the unlimited began to be drawn in and limited by the limit.” He further implied that the Pythagorean teaching of the limit and no-limit were direct results of the philosopher Anaximander who argued “(that which is) unlimited”, “boundless”, “infinite”, or “indefinite” as “Apeiron” and “peras” as “end, limit, boundary”.
Modern philosophical instinct and training implies the definition of space as limit and no-limit in dual linear and circular terms questionable considering one is presented with two perspectives: They are an empirical contradiction or a transcendental paradox in the respect that logic either nullifies itself or transcends pasts its origins. (quote) Neither school of thought gives any real justification as they manifest a dualism in which one perspective attempts to wrestle over the other, resulting in a Neitschian view of force embodied as “perspectivism” , Pythagorean definition where duality is conduce to change, or the problem of Wittgenstein where “[a]ll the propositions of logic are generalizations of tautologies and all generalizations of tautologies are generalizations of logic. There are no other logical propositions.”
Paradoxically, the western empirical linearism and eastern transcendental circular forms of logic, both need eachother as one exists as the “limit” which defines the other. The western laws of logic observe the “fallacy of circularity” as a justification for linearism. The eastern views observe the deficiencies of individuative linearism promote holistic circularity. In a separate respect both observe a nature of “no-limit” through western regressive and progressive rationality dependent to a degree on infinitism and eastern circularity dependent on holisitic centering and rotation as absent of limit.
The problem occurs in the respect that we are constantly limited to dualisms, and the problems of logic and definition are reduced to ones of dimension. These dualisms create a problem of definition dependent upon polarity, observed in the hermetic philosophy as the “Principle of Polarity”. One polarity defines the other while simultaneously causing a perpetual sense of definition between the two. Polarity can be viewed as a contradiction of force under the Nietzschean metaphors of Apollo and Dionysus , an absence of structure as the Pythagorean Dyad , and an alternation of definition through the hermetic “Principle of Frequency” . Or it may simply just be observed as a problem in the same manner of Cartesian Dualism and Platonic Dualism leading to the competing substance, property, and predicate dualistic perspectives that provides for the universal means of division in philosophy between materialism and idealism .
A third more rational approach must be taken in order to deal with the multiple dualisms inherent through the limit and no-limit definitive nature of space in both form and function, quantity and quality, circularity and linearism, and western empirical and eastern transcendental logic. Without a solution to these reoccurring duals a process of fracturing takes place in which each definition is dependent on an infinite linear regression, circular justification or simply an acceptance of the axioms without any observable definition, all of which are observed in the Munchausen Trilemma .
This fracturing can be implied as a form of logistic Atomism, observed by the more modern philosophers Russel, Wittgenstein and Carnap and stemming from the pre-Socratic philosophers Leucippus and Democritus of Abdera , in which the linear regressive separation, circular definition, and axioms can be observed as individual units in themselves that must relate through a process of continual change. Hence it may be implied that dualisms are dependent on relativistic logistic unit-particles that exist through continual individuation as a form of definition.
The problem occurs in the respect that the very problem of definition we seek to avoid, change, appears to be one of the very foundations for this very act of definition. We can simply observe this as relativity merely being individuative limits and no-limits that exist as spatial dimensions of change as unit-particulate. This provides the foundation for not only materialistic change as a refutation of idealism observed by Kant , further evidenced by physics dependence on Principle of Locality in the Principle of Relativity, but further change in abstract logistic structures such as moral relativism, truth and cultural relativism , methodological relativism, relationalism and dialectic materialism . However, the observation of change as a constant, through logisitic unit-particulate, observes a consistency which further results in another dualism of change and no-change, which is further exacerbated by the number of aforementioned dualities premised in spatial limit and no-limit.
The question is what is the unifying median of all these definitions? Again intuition and reason leads one to the first axiom of geometry: “the point” which is again inherent within the question itself. Space as a unifying medial point which acts as a common bond gives a starting definition as to the nature of space itself. And what is this starting definition? Space begins with the “point”. Viewing this foundation of space through the standard Euclidian definition, or even non-Euclidian, as a 0d entity would be instinctive from the standard academic school of thought, however a few problems occur based upon this simple premise.
Our observation of the 0d point is strictly the observation of an individuator which divides and multiplies 1d lines, at minimum. It observes a degree of unity as “part”, “particulate” or “unit” but not “unity” itself. This is considering the 0d point lacks dimensionality and depends on the dimensionality of the 1d line ??and other dimensional spatial structures?? in order to be observed. Absent of the dimension, the 0d point acts as an individuator which relates through 1d lines. These 1d lines exist as extradimensional entities, which continually project past their origins of the 0d point. However when strictly relative to the 0d point, they project nowhere as observed ((((in the Aristotelian argument of movement in a void))). The 1d line must relate to other 1d lines, through the 0d point, in an equal manner to which a unit (or particulate) must relate to further units (or particulate in order to exist. In these respects, the 0d point as an individuator of units or particulate is dependent upon an extradimensionality which projects pasts its origins. In these regards extradimensionality is synonymous to individuation as relation through 0d.
So what to do? How can a unity be observed through space when the 0d point/1d line dualism itself cannot be a “unity”, but rather a “unit”, because of it’s inherent Extradimensional nature being a cause of individuation. We see this intuitively within various cultures which used and still use the line as a marker for “unit.”
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 identified. 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.