TRINPsite 55.04.2 - 55.04.2  
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So far as catenas of the zero- and first levels of reiteration are concerned, the following catenas turn out to be the interesting ones from the neutralist perspective:

  • (a)  basic catenas;
  • (b)  bicatenally derived, bivariant difference catenas;
  • (c)  differentiation and time-differential catenas conditionally;
  • (d)  other differential-catenas and quotient-, integral- and product-catenas.
The norm of neutrality does not discriminate between the material and the nonmaterial world, or between spatiotemporal and non-spatiotemporal reality in any way. Hence, it is merely to illustrate the basic meaning of the norm of neutrality that we will briefly consider here what is superior or inferior according to this norm with regard to a number of simple, spatiotemporal dimensions. Since this illustration cannot become too technical, we must leave out quotient-, integral- and product-catenas, and thus we will not be discussing the exact derivation of, say, the smallness catenas, altho they are associated with simple dimensions like surface area and volume.

Since the principle of neutrality does not apply to time, or a connected series of time predicates, the basic catenas to which it does apply are related to three or more spatial dimensions. We will confine ourselves to one spatial dimension and will term the catena associated with it "the longitude catena" (cp. 'latitude' and 'altitude catena' for a second and third, spatial dimension). Assuming that there is such a catena is tantamount to assuming that there is a neutral longitude between positive longitudes on the one hand, and negative longitudes on the other. But obviously, there is no fixed 'neutral' point; conceptually speaking, we could choose any, provided that it is not at the 'end' of the universe. And so far as we know, there is no empirically given neutral longitude either. Many theorists will, perhaps, be eager to point out that time and space are relative notions, but the matter is not that simple. For if relative is supposed to mean something like comparative and difference-catenary, this presupposes the existence of an original catena of the difference catena in question; let us say, at least in a conceptual sense. Furthermore, to say that relative means relational will not help very much either, because having a one-place predicate of a basic, spatial catena can be construed as having a two-place relation with a thing at a hypothetically neutral point.

Even if it is true that the neutral longitude is superior to any other longitude, this seems to have no practical significance. On the hypothesis of mean-neutrality the neutral longitude is the mean longitude of all longitude-catenals, that is, not just the longitude-catenals in the 'universe' we happen to live in, but in all 'universes'. (And on the average all longitude-catenals in whatever 'universe' have the same, neutral position.) This plainly does not give us an empirical clue either. However, if there were such a given, neutral point, it would be better for a material object to be at this point. The farther away a spatiotemporal thing would be from this point, the worse it would be, normatively speaking and all other things being equal.

For the same reason as it is arbitrary to fix a so-called 'universally neutral' longitude-catenary point, it is arbitrary to fix any point, and therefore we need not discuss the bicatenal monovariant positivity-difference catena of the longitude catena at all, tho this catena is not factitious. The first result (and a spectacular one) does not come in until we start to consider the bicatenal bivariant difference catena of the longitude catena. In the context of this catena a difference in longitude is either positive or negative, and having the same longitude is neutral. According to the principle of neutrality longitudinal equality is therefore superior to longitudinal inequality, whether positive or negative. And the same holds for the other spatial dimensions. Any force aimed at spatial equality, that is, at having the same spatial position, is therefore neutral-directed. One such neutral-directed force is in fact the dominating force in the universe at large. It is called "gravitation". It is only claimed here that gravitation is a neutral-directed force with respect to the bicatenal bivariant difference catenas of the spatial catenas. The reverse is not true, namely that any such neutral-directed force could not be anything else than gravitation, for the nuclear force or 'strong interaction' which holds protons and neutrons together in the subatomic world is a similar kind of force. But why was gravitation not mentioned before as one of the great neutral-directed forces of nature among other examples of striving for neutrality? The reason is that on the face of it the force of gravity, and also the nuclear force, seem to be forces aimed at extremity (at extreme proximity, to be precise) and it needs a little bit of catenical analysis to show that this view is mistaken.

Neither the positive extremity of the proximity catena nor the fact that a force like gravitation is manifested by acceleration determines its being neutral-directed, or not, in terms of the norm of neutrality. The proximity catena is a modulus-catena of the bicatenally derived, bivariant positivity-difference catena of the longitude catena (and similar, spatial catenas). It is therefore a factitious catena. Nonfactitious is the original catena: the bicatenally derived, bivariant difference catena itself. And it is with respect to this catena that a force is neutral-directed or not. What has happened is that the distinction between a positive difference in longitude and a negative difference in longitude was not believed important (and rightly so), and that any such difference has been called "distance". Since this 'distance' is usually by definition positive, having-no-distance or being-at-exactly-the-same-point came to be thought of not only as a limiting case (which is correct), but also as an extreme case (which is fallacious in terms of the original catena). Distance always being positive, and having-no-distance merely being a limiting case, people started talking about "large" and "small distances", thus introducing a new catena of predicates: proximity (corresponding to a small distance), the vague being-neither-close-nor-far (a distance which is neither small nor large) and farness (a large distance). The principle of neutrality, however, does not apply to this modulus-catena, because original catenality takes precedence over derivative catenality. (Why it does not apply to acceleration either, we will discuss shortly.)

Life on Earth, or on any other planet, is unthinkable without the mutual attraction between this planet's mass and bodies at or near its surface, or between material entities (such as bodies, particles and quanta) in general. The scope of this neutral-directed attraction is universal. It is present on the level of galaxies and on the level of our daily life (as the force of gravity) as well as on the level of the subatomic world (as the nuclear force). The force of gravity and the nuclear force themselves are not normatively superior; instead it is their being directed at what is normatively superior which should impress us. Now, skepticists may easily come up with examples of particularly gravitational effects which are nothing to be joyful about. They may point at children or people falling into deep ravines or out of windows of tall buildings and being crushed to death because of the earth's attraction. (Yes, Mother Earth's love can be rather ponderous.) Apart from the fact that they then put all emphasis on a few exceptional cases, they do not seldom confuse people or mental beings and material bodies. In the strict sense, gravitation is a neutral-directed force which does not affect mental beings at all, but only bodies, including those of people. Furthermore, a body may have all kinds of other properties which are dramatically changed when it hits the earth, yet the question of the normative evaluation of those changes is a different question altogether. People or children, too, may be in agony when they fall, or have fallen, from a great height. Nevertheless this great unhappiness is not what gravitation is about. Gravitation is a force operating in the spatiotemporal field, and to say that it is neutral-directed is to say that it aims at neutrality in this field, a neutrality which is normatively superior, all other things being equal. A world governed by gravitation may only be compared with a world without gravitation when the latter world is equally happy, equally unhappy, or also neither happy nor unhappy.

Where neutrality needs to be restored, established or maintained, this requires certain kinds of polarities of the differentiation and time-differential catenas. Differentiation with respect to the basic longitude catena is a change of longitude, time-differential bipolarity is movement. But this change and this movement is change and movement with respect to an (imaginary?) neutral point or a body located at this point; it is not change and movement with respect to an arbitrary other body. The latter predicates do not belong to the differentiation and time-differential catenas of the basic catena but of its bivariant difference catena. It is positive neutrality-differentiation and time-differential catenality which is needed to further the neutrality of the original catena. The neutralities of these catenas are constancy and rest. They are superior, unless positivity, that is, a change or movement in the direction of the neutrality of the original catena, is needed to promote the cause of original neutralness. Negativity, that is, a change or movement in a direction away from the original catena's neutrality, is normatively inferior in all instances. Whether positive or negative, change and movement can never be ends in themselves on the norm of neutrality; if allowable, they must always serve a more urgent, neutral end. On the other hand, constancy and rest may be taken as ends in themselves, unless a more urgent, neutral end requires change and movement. (What is more urgent follows, firstly, from the position of the neutrality in the derivation system concerned, and secondly, from the relative weight of neutralities belonging to different derivation systems, an issue to be discussed later.)

When the distance between two objects becomes smaller and smaller (and their difference-catenary value approaches 0), we say that they move with respect to each other, or simplifyingly, that the one object is at rest, while the other moves towards it (particularly when this latter object is much smaller and lighter). This movement itself tho can theoretically be a constant movement, when the velocity is uniform, or a changing one, when the velocity is not uniform. Not only can the movement or velocity itself be directed towards the neutrality of the original catena, also the change of movement or velocity can. In the case of positive neutrality-differentiation it is positive positivity-differentiation which promotes the original neutrality. Positive neutrality-differentiation would then work against any change in velocity which would bring the original end of neutrality nearer. Hence, the type of acceleration manifested by the force of gravity is the positivity of the time-positivity-differential catena of the time-neutrality-differential catena of the spatial bicatenal bivariant difference catenas. (Note that this positivity does not represent the 'essence' of acceleration. 'Acceleration' is, properly speaking, increase of velocity regardless of the object's position and direction, and thus it would be the negativity of the time-positivity-differential catena of the modulus-catena of the neutrality-differential catena of the bicatenal bivariant difference catenas. This should demonstrate how simple mathematical-physical quantities only show the surface structure of catenary reality.)

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Model of Neutral-Inclusivity
Book of Fundamentals
The Norm of Neutrality