On Communication

Communication is meeting of Languages. I use the word Language to express anything we see. Be it a rigid structure or some fluid or a usual language.

Identity

For once,

  • Let there be a Identity Language.

  • Let there be a complementary pair of Languages formed from the Identity Language.

Hypothesis:

  • Using the above to statements repeatedly (complementarily), we may generate a Language.

  • Conversely, a given Language may be Identified to the Identity Language.

Identification is both ways (complementary). Hence, we may start from any one suitable Language and Identify the others with it. The repitition of steps involved may be finite or infinite depending on the Language.

Complementary Pairs of Language

First we shall show the converse: Given a Language, we will Identify it to the Identity Language. Then we shall do it the other way round. When the cycle is complete, we will see that any given Language could have been used as the Identity Language. To simplify the process, we will use symbols. For the Identity Language, we shall use a circle: \(\circ\) as the symbol. For the Complementary Pairs of Languages which form the Identity Language, we shall use two complementary pairs of the circle \(\circ\). Each complementary pair misses one point on the circle. The corresponding complementary points are expressed by \(-1\) and \(1+\). We may name the complementary pairs corresponding to the missing points to be the complementary symbols \(-\,\,-\,\,1\) and \(\,\,-\,\,1\,\,+\,\,\) respectively. To simplify, the complementary pairs of the circle are expressed by \(-1\) and \(1+\) and the complementary points are expressed by \(1+\) and \(-1\) respectively.

  • The simplest case is: Given two complementary pairs of Languages, how to identify them to Identity Language?

For the Complementary Pairs of Languages which form the Identity Language, we shall use two complementary halves of the circle \(\circ\) expressed by \(-1\) and \(1+\). Positioning the complementary halves \(-1\) and \(1+\) together by placing both on top of each other, we see that the complementary halves provide each other their missing point (that is, \(1+\) and \(-1\) respectively) which results in a complete circle \(\circ\). This is seen in the expression \((-\,\,1 \,\,\,\,1\,\,+)(-\,\,1 \,\,\,\,1\,\,+)\). In Mathematical Language, the algebraic simplification of this expression is done by identifying together complementary pairs of symbols by an Identity symbol (the Identity Mathematical Language denoted \(0\)). This results in the following steps:

  • \[(-\,\,1 \,\,\,\,1\,\,+)(-\,\,1 \,\,\,\,1\,\,+)\]
  • \[(\,\,0\,\,)(\,\,0\,\,)\]
  • \[(0)\]
  • \[0\]
  • .

We could have used another way. This is by naming the Identity Language as \(1\). This works within the level of the given symbols for their Identification as seen in following steps:

  • \[(-\,\,1 \,\,\,\,1\,\,+)(-\,\,1 \,\,\,\,1\,\,+)\]
  • \[(\,\,1\,\,)(\,\,1\,\,)\]
  • \[(1)\]
  • \[1\]

Alternatively, the symbols could have been interchanged to see the following steps:

\[-\,\,1 \,\,(\,\,-\,\,1 \,\,1\,\,+\,\,)\,\,1\,\,+\]
  • \[-\,\,1 \,\,(\,\,1 \,\,)\,\,1\,\,+\]
  • \[-\,\,1 \,\,1 \,\,1\,\,+\]
  • \[(-\,\,1 \,\, \,\,1\,\,+)\]
  • \[-\,\,1 1\,\,+\]
  • \[1\]

Or, using \(0\) as the Identity symbol,

  • \[-\,\,1 \,\,(\,\,-\,\,1 \,\,1\,\,+\,\,)\,\,1\,\,+\]
  • \[-\,\,1 \,\,(\,\,0 \,\,)\,\,1\,\,+\]
  • \[-\,\,1 \,\,0 \,\,1\,\,+\]
  • \[(-\,\,1 \,\,0 \,\,1\,\,+)\]
  • \[-\,\,1 1\,\,+\]
  • \[0\]
  • .

Or, the symbols could have been Identified in another way:

  • \[(-\,\,1 \,\,-\,\,1\,\,) \,\,(\,\,1\,\,+\,\,1\,\,+\,\,)\]
  • \[(\,\,1 \,\,)\,\,(\,\,1 \,\,)\]
  • \[1\]

Or, we may use \(0\) as the Identity symbol:

  • \[(-\,\,1 \,\,-\,\,1\,\,) \,\,(\,\,1\,\,+\,\,1\,\,+\,\,)\]
  • \[(\,\,0 \,\,)\,\,(\,\,0 \,\,)\]
  • \[0\]
  • .

Thus, we arrive to see Identity Language.

Complementary Pairing of Pairs of Languages

Complementary pairing of Languages Identifies the Languages to an Identity Language. However we see that this involved, creating another complementary pair of points corrseponding to the complementary pair of Languages. These complementary pairs of points may themselves be seen to be complementary Languages. Thus, a Pair of Languages which Identifies to the Identity Language also has its corresponding complementary Pair of Languages which get Identified to the Identity Language in the process of Identifying the first pair. In Mathematical Language, this may be seen as a cyclic group of order \(4\). Thus, the previous process is equally valid to show how to Identify a group of 4 Languages with two pairs of complementary pairs.

Repeating the above process of Identifying, we see that all Languages corresponding to elements in a cyclic group of order \(2^n\) are Iedntified to the Identity Language by Identifying one complementary pair at a time.

Identifying that pairs of complementary pairs may be Identified one at a time, a cyclic group of order \(4^n\) are Identified to the Identity Language.