Abstract

Giving an Identity is naming. Language is positioning together Complementry Pairs of Identities. Classifying Languages is Classifying how to position Complementary Pairs of Identities. In the finite case, this is reduced to the corresponding work done in Mathematics Language: Classification of Finite Simple Groups. In this paper, I will focus on Cyclic Groups.

Identity

Identity is name. Unique Identity is Unique Name.

Identification

Naming is Identification. In the simple case: Identities come in Complementary Pairs. Every Complementary Pair itself Identifies to give a Unique Identity. Positioning together Complementary Pairs is Identification.

Whenever there are a variety of names (identities), they can be placed together and given a unique name. Thus, Identification of individual Identities is the same as giving them a unique name (Unique Identity). For example:

  • In schools, Identification is what we do when we bring together a variety of individual students with their own names (individual identities) in a class. This process of giving a common name (Identity) to a group of students in the form of a ‘Class’ is an example of Identification.

  • In schools, Identification at a larger scale is what we do when we make individual students wear a common uniform. The uniform is the unique Identity of all students enrolled in the school.

  • In schools, Identification at an even larger scale is what we do when we give the Name of the school. The name of the School is what Identifies all students of a school from different years.

Identities in Complementary Pairs

Thus, we see that whenever there is a variety of Individual Identities, placing them together to give a Unique Identity creates a Unique Name (Identity). The Unique Identity may be seen to either be at the same degree or a higher degree than that of the Individual Identites or at a lower degree than the Individual Identities. That is, the Unique Identity obtained by Identification of individual elements is the complete cycle of which, the Individual Identities are the varieties. For example:

  • In schools, Class is all students put together. The students in a Class are Individual Identities which together form the Class. Hence the Class itself is seen at higher degree than the individual students. The other way round, the Class itself is formed by the Identification of all students put together. Hence, the Class gets Identified by placing together the Individual Identities (names) of all students in the class. In this way, Identifying a Class is placing together a variety of Individual Names. Thus, the Class is seen as a Unique Identity at lower degree than that of the Individual Identities (since Class is one name which is lesser than the all number of names). A particular student who is representative of the Class may also act as the Unique Identity of the Class.

  • In schools, Uniform Identifies students in all Classes. The Uniform is placed with every student and hence it doesn’t distinguish any Individual Identity (student). So, the Individual Identity of any student is not affected by placing it together with the Unique Identity (uniform). However, by seeing the Uniform, every student sees that they have a Unique Identity. Thus the students who want to connect with each other (Identify with each other), may look at the Unique Identity instead of looking at the Individual Identities as a common base to Identify with another student. The uniform may not help at the level of individual interaction, but it serves as a common base to connect together students. This shows how the Uniform (Unique Identity) is at the same degree as that of its Parts (Individual Idnetities). It can be placed with the parts but does not affect their Individual Identity at all. However, it connects together the Individual Identities by forming a base (Unique Identity). The base (Unique Identity) may be seen at a lower degree than the Individual Identities by considering the Individual Identities to be additional to the Base. The other way round, the Unique Identity (Base) may be seen at a higher degree by identifying the differences of Individual Identities. This is done by looking at all of the Individual Identities placed together with the Uniform to give a continuously extended version of the Uniform.

  • In schools, Name of the school Identifies students who were part of the school over time. Thus, the Name Identifies students on a Higher Degree. On the other hand, every student representing the School, identifies the Name of the school when they represent the school in Inter-School tournaments. Thus, the Name of the School may be identified at the Same Degree as that of a student. On the other hand, the Name of the School is an Identity formed by placing together all names of students. Hence, the Unique Identity is lower in degree than the Number of names of individual students (Individual Identities).

We see through the above that Identification is Classification. Unique Identity is at the same degree or higher degree or lower degree. We see that every time there is one way to see the Identity at Higher Degree, there is another way to see the Identity at Lower Degree. Thus Unique Identity at Higher and Lower Degree are complementary pairs. The Unique Identity at the same degree is the Identification of both these pairs. Thus, we shall formalize this notion in condensed form by using some Mathematical Language.

Complementary Pairs form a Cycle

A cycle is formed by balancing two sides. We name the two sides to be complementary pairs. Thus, placing together (Identifying) complementary pairs forms a cycle. To express that the cycle contains both sides, we say that a cycle is conserved. We may also express this by saying that a cycle is completion of the two complementary pairs. That is the cycle is a Unique Identity of the two complementary pairs.

Classifying Languages

We see that giving a unique Identity to bring together Individual Identities is what is done in a Class. Hence we may see Identification as Classification. I started to classify Languages and realized that the result will match what has already been done in the Language of Mathematical Groups. This is seen by my formulation of Languages to emerge in complementary pairs (element and its inverse to form identity). Thus, Languages can be classified corresponding to the classification of finite simple groups. Here is the complete list Classification of finite simple groups. Right now I focus on Cyclic Groups. I name the Languages corresponding to Cyclic Groups as Cyclic Languages. At some point, I will only focus on Identity Language (Cyclic group with one element as identity). As an intermediate step, I will see the Cyclic group with two elements (Cyclic group of order \(2\)).

Degree of a Cyclic Language

Degree of a Cyclic Language is the Order of its corresponding Cyclic Group.

Identity Language

Identity Language corresponds to the Group with one element: Identity. There are many ways to image-ine this. Some examples are:

  • A Blank white space.

  • A Letter \(I\).

  • A letter \(e\).

  • A circle \(\circ\)

  • A unique cycle.

  • A sphere.

  • A cone.

  • A square

While all of the above are equaivalent, for simplicity, I choose to picture this as the symbol \(\circ\) which shows a cycle.

  • Observe that the Identity Language can be reinterpreted to be a Group with any number of elements with all of them being the Identity element. This gives an isomorphism to Cyclic groups of order which is equal to the number of elements. For example:

    • Identity element positioned with any element gives the element itself. Hence one may write \(\,\,E\,\, \cdot \,\, E\,\, =\,\, E\,\,\). Thus, we may think of this as a cyclic group with two elements, both being Identity. Here each element is its own inverse. One may give individual names (complementary identities) to the elements to create variety and see that when one element is distinguished the other element becomes its inverse. This may be seen by writing the Group as \(\{E, I\}\) and \(I \cdot I = E\). This is isomorphic to the cyclic group of order \(2\).

    • Similarly, we may write \(\,\,E\,\, \cdot \,\,E\,\, \cdot \,\,E\,\,= \,\,E\,\,\) and think of a group with three elements, each being Identity. The element \(E\) has \(E\,\,\cdot\,\, E\) as its inverse and vice versa. One may rename these elements to create a variety and have an isomorphism to a cyclic group of order \(3\). That is, the group becomes \(\{I, I\cdot I, E\}\) or \(\{I, I^2, E\}\).

Once upon a time, there was a circle \(\circ\)

Every point on the circle \(\circ\) is same when viewed from the center. That is, we see a difference in two points because of our position of viewing them. In positioning ourselves equally from all points, we identify all of them to be the same. The other way round, this can be seen as the points identifying to maintain equal position with respect to the center as their Identity. Thus, all points at equal distance from the center are Identified by the center and all points at equal distance from the center identify to the center.

  • Identifying the center of a circle to be a point.

    • We rewrite the statement as: All points at equal distance from ‘the’ point are Identified by ‘the’ point and all points at equal distance from ‘the’ point identify to ‘the’ point.

  • Identifying ‘by’ and ‘to’.

    • We rewrite the statement as: All points at equal distance from ‘the’ point Identify ‘the’ point and all points at equal distance from ‘the’ point identify ‘the’ point.

  • Identifying ‘all points’ as ‘the point’

    • We rewrite the statement as: the point at equal distance from the point identify the point and the point at equal distance from the point identify the point.

  • Identifying ‘the point’ as ‘the point’

    • We rewrite the statement as: the point at its own distance identifies the point at its own distance.

  • Identifying ‘own distance’as ‘the point’ itself

    • We rewrite the statement as: the point identifies the point

    • We rewrite the statement as: the point is its own identity.

  • Identifying ‘the point’ as ‘identity’

    • We rewrite the statement as: the point is.

    • We rewrite the statement as: the point.

    • We rewrite the statement as: point.

    • We rewrite the statement as: .

    • We rewrite the statement as:

This shows how given any Language, we can Identify it to the Identity Language. Simply, keep identifying complementary pairs. The reverse is also true. One may start with Identity Language and keep positioning complementary pairs which identify to generate a Language. The naming doesn’t matter for the end. We could have identified all points as centres and the last statements would have read as:

  • We rewrite the statement as: the center is its own Identity.

  • We rewrite the statement as: the center is.

  • We rewrite the statement as: the center.

  • We rewrite the statement as: center.

  • We rewrite the statement as: .

  • We rewrite the statement as:

Another way is to identify ‘the center’ to ‘is’:

  • We rewrite the statement as: the center is.

  • We rewrite the statement as: is.

  • We rewrite the statement as: .

  • We rewrite the statement as:

So we have seen that the story “Once upon a time there was a circle” is coming from the story “Once upon a time there was a point”. A point is the circle. This serves to show that any Identity gives rise to its complementary Identity. These two composed together form another Identity. This new Identity again gives rise to its complementary Identity. It is our choice to see all as one single Idnetity or as a compositioning of multiple Identities. All points of view are equivalent.

Complementary Pairs as Identity Language (Cyclic Group of Order 2)

Instead of seeing a unique symbol, we may see a cyclic Language comprised of two complementary pairs to be the representing the Identity Language. Since Languages come in pairs, in any image, we have meeting of two complementary languages. This comes from the identification of a pair of Languages to be a cycle. For example, the blank white space we imagine is always bounded by a less white boundary. A letter has its background as its complementary pair. Any symbol has its background as its complementary pair. The two together form a Higher Cyclic Identity and the two Identified at their Boundary form a Lower Cyclic Identity. For example:

  • Second Roots of Unity: \(-1\) and \(1+\) are two points on the Axis which form a complementary pair of Identities. These Identify together to form \(S^0\) (I prefer to call it \(S^{\frac{1}{2}}\)) as the Identity. These Identify at a Lower Degree as the Identity \(1\), which is the Origin. When these two points are connected (Identified) by the semicircles of the unit Circle they form \(S^1\) as the Higher Degree Identity.

  • All the “we rewrite” statements above can be read in reverse order to see how to start from an Identity and proceed to higher and higher degrees of Identities.