This document continues the investigation of categories and complexity layers.

Ontological Status of Whatness Categories

We will now discuss **whatness categories** as to their way of being.

Conventionally the whatness of something is expressed as a *definition.* In it the **meaning** of some concept is given. As such a *definition* is a logical or gnoseological entity, not an ontological entity, and thus not a category or If / Then constant, because the reference is to *concepts,* not to things, materials or their properties. The direction of 'movement' is from unanalysed (not understood) to analysed (and therefore understood), for instance in

*Man is a featherless biped.*

In a **whatness category,** on the other hand, the direction is from analysed to unanalysed. And here we have, moreover not to do with concepts and their meaning, but with beings (which can be either physical or mathematical), materials and properties .

The whatness category (eidetical constant) can first of all determine the intrinsic whatness of a genuine being (When this being is mathematical there is no difference between intrinsic and extrinsic whatness). In the physical case such a whatness category or eidetical constant is the dynamical law of the physical dynamical system that generated that being ( This dynamical law is at the same time a nexus category, connecting states of the dynamical system).

The whatness category can also determine the intrinsic whatness of some given material, and, finally it can determine the whatness of some property of a given being or of a given material.

In the case of a whatness category, the "*unanalysed*" does not refer to something-not-understood, but to something-being-just-there. And the latter __is__ *necessarily* what it intrinsically is, i.e. it is determined as to what it intrinsically is by the "*analysed*", where the latter does not mean analysed by __us__. It means the ontological 'machinery' that determines something as to what it is in itself.

As an example of a whatness category we take the **Category of 3-fold Cyclic symmetry** that was discussed **earlier** ( Part I ).

**Category of 3-fold Cyclic Symmetry** (whatness category) **:**

Let us analyse this particular whatness category.Ifall cover-operations (let us call them)1, p, pof something form the group (as given by a group table)^{2}:

thenthat something has3-fold cyclic symmetry.

It is supposed to determine

From this table we see that the following relations between the elements **a, b** and **c** hold **:**

ab = b

ac = c

ba = b

bb = b

bc = a

ca = c

cb = a

cc = c

These relations show that the element **a** is the identity element of the group. It does not bring about a change when combined either with itself or with any other element of the group. And, again, what the other elements, viz. **b** and **c** , are, is laid down completely and exclusively by their relations to the other elements, where these relations are given in the group table. There is no object (neither a mathematical object, nor a physical object (thing)) involved in this abstract pattern. The latter is just an algebraic pattern.

So with this algebraic pattern we have now established the kernel of the whatness category of 3-fold cyclic symmmetry. This kernel must now be 'upholstered' in order to arrive at symmetry. To begin with we must interpret the group elements as *cover-operations.* A cover-operation is a transformation of an object (mathematical or physical) such that the result is precisely superposed upon the object as it was in its initial state. So now at least some *object* is involved. But this object is still not the (3-fold) symmetry. To involve cover-operations we can interpret the group elements **a, b, c** accordingly. First we saw that **bb = b ^{2} = c** . So we can express

If we interpret these elements **1, b** and **b ^{2}** as cover-operations, then (I) the element

The group elements (which constitute the above mentioned algebraic pattern) are now cover-operations, and that means that they are **symmetry transformations.** But as such they are still abstract, in the sense that they are transformations *of something.* And only the latter, or its parts, are *beings.* So we must see what we can do with these (still abstract) group elements. See next Figure.

In the above Figure we see a two-dimensional object (and thus surely just a mathematical object), consisting of three parts **X, Y** and **Z** , that will be superimposed upon itself when it is rotated either 0^{0} or 120^{0} (anticlockwise) or 240^{0} about an axis perpendicular to its plane and going through its center. In fact what we see, when performing these rotations, that the parts **X, Y** and **Z** of this object are transduced into each other by the symmmetry transformations **1, p** and **p ^{2}** according to the following scheme

The above shows that we can identify part **X** of the object with group element **1** , part **Y** with the group element **p** and part **Z** with the group element **p ^{2}** . So now the group elements of our group not only signify symmetry transformations, but can also be represented by

Figure 1. *The group ***C _{3}**

Figure 1a. *Construction of the gyroid pyramid from the gyroid polygon of the previous Figure. Black lines belong to the pyramid's base, while the red lines are going up, i.e. leave the plane of the base.*

The next two Figures give the finished construction of the gyroid pyramid. It has the same symmetry -- C

Figure 1b. *Three-fold gyroid pyramid, constructed from its base (Figure 1 ). See also next Figure.*

Figure 1c. *Same as previous Figure. Visible areas colored. The pyramid consists of three parts that can represent the three elements ***1, p, p ^{2}**

Every coherent object in the Physical Layer possesses a specific

Indeed, in all this, it is just

The most convincing argument that genuine objects do have a

Another argument is the fact that

So despite the all-permeating r e l a t i o n a l i t y , which -- as

So ultimately there must be things (objects) that have their identity not radiating out of it, but have it within. And this means that they are

The expression of whatness categories in the form of If / Then

So (to take an example)

The next Figure illustrates how objects are connected to each other by entitatively determinative threads.

Figure 2. *Physical objects -- symbolized as discs, triangles, polygons, rectangles, etc. -- connected by entitative threads. The red dots in the objects symbolize their ESSENCE, and from the latter the intrinsic properties of the relevant object are phenotypically expressed. See also ***Figure 13*** of the previous document.*

Let us explain the above Figure.

The objects

In the same way the objects

Further, the objects

And also the objects

And, finally, the objects

So the object

Let these five properties, that are intrinsic to

The above drawing is about the **ontological constitution** of things. As such this is a static state of affairs. Now we want to involve **nexus categories** in that drawing, that is to say we now are going to add a dynamical element.

A nexus category determines the necessary connection between states of a dynamical system. When we consider only two consecutive states, we have -- in the Physical Layer -- to do with causality. If we consider all possible sequences of states, going out from all possible initial states (initial conditions), then we have to do with a certain special nexus category in the form of a dynamical law (which presupposes causality).

A part of a dynamical trajectory (i.e. a part of a sequence of states of the dynamical system) can be exemplified by a series of larval stages during the individual development of some insect. When we concentrate on two consecutive larval stages, Stage n and Stage n+1, we can say that the former is the cause of the latter.

What we see when such an individual insect develops, is in fact the **successive phenotypical expression and suppression of intrinsic properties by the insect's ESSENCE** (i.e. some properties appear first, and later on some other properties, while some disappear again, etc. In all this the insect's ESSENCE remains the same.).

The next diagram repeats the previous one and shows where a **nexus** fits in, a nexus, between a *concretum* and a(nother) *concretum*, determined by a (special) nexus category.

Figure 3. *Insertion of a ***n e x u s*** into the previous diagram (Figure 2). The objects ***k _{n}**

The nexus is indicated by a blue arrow. It is determined by a nexus category (presupposing causality), which is the dynamical law of the dynamical system of which

Until now we have discussed mainly some

For such categories HARTMANN has established inter-categorical laws. These laws in fact describe positively what such categories really are and how they are related to each other.

Before we look into those categories and the corresponding inter-categorical laws, we will first continue to concentrate on special categories in order to see them in the context of a definite research program, which will later include the mentioned (more) general categories. Our

In the

A great many documents will follow, all concentrating on just a few (but interesting!) special categories, namely *Intrinsic Shape, Symmetry* and *Promorph*, accompanied with some ontological discussions.

But if the reader is more interested in an **overall view** of the *Ontological Structure of the World* in terms of the Layer Theory, categories and categorical laws, he or she can skip all the documents about shape, symmetry and promorph, and can pick up the theoretical discussion again at **Part XXIX Sequel-24** . There the Theory of Ontological Layers is fully developed, and will form the philosophical context of an extensively worked-out Crystal Analogy.

If, however, the reader wishes to continue the study of the (special) categories *Intrinsic Shape, Symmetry* and *Promorph*, he or she should use the LINK directly below.

To continue click HERE

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