(This is incomplete and currently being updated regularly, please be patient for the final version)

Duality

This section can be read without trying to make much sense of it. At the end, one can come back and go through it again.

  • Identity is.

  • Identity is not.

  • Neither of the statements above make any difference since Identity is and is not. Identity is Everywhere and Nowhere. One can proceed with either statement or no statement at all and it makes no difference, as we shall see later. Since existence or non-existence of Identity does not make any difference, please allow me to make the following statements about Identity which amount to nothing.

  • Identity cannot be captured entirely and yet all we end up doing is capture Identity entirely.

  • Identity is Dual to Duality.

  • Identity is a Differential form of Duality and Duality is an Integral Form of Identity.

  • Duality is Creating and Destroying Boundaries to Partition the Identity.

  • Every Language (Structure) is based on Duality.

  • Boundaries and their corresponding Dual Phases are the Language (and that’s all there is to a Language).

  • Boundaries

  • A Boundary and its corresponding Dual Phases together form a Composition.

Language

I would like to use the word Language to be any structure used to represent knowledge. As in usual usage, when both language and knowledge are specified, a particular language will be used to represent the specific knowledge. To clarify again: whenever I use the word ‘Language’ with a capital L, I mean anything that we use to represent knowledge. I loosely refer to anything specific (already represened in some Language) that we would like to represent (in another Language) as ‘knowledge’, with a small k.

Dualities are Fundamental Laws of a Language - Every Fundamental Law is an Equilibrium Phase Diagram

  • Duality is Differentiation and Integration.

  • The Fundamental Law of Degree \(0\) is the Identity of Composition.

  • Every Fundamental Law is a Composition Law of Duality. This means that every Fundamental Law is obtained by Composing Dualities. This is just a restatement of the fact that every Fundamental Law is an Equilibrium Phase Boundary. In particular, every Mathematically correct Equation is an Equilibrium Phase Boundary which can be obtained by composing Dualities.

  • This is why Fundamental Laws are Natural: They Appear and Disappear naturally at the same time as expressions of Duality.

  • Every Equilibrium Phase Diagram is a Fundamental Law and all that is a Fundamental Law is a Phase Diagram. The Universe itslef is a Phase in a Phase Diagram. On the other side of the Equilibrium Phase boundary of our Universe is our Dual Universe. Differentiation and Integrations connects all Phases to one another and hence allows us to go to Higher Degrees of Freedom.

  • The Fundamental Laws themselves have Degree of Freedom lesser than the Degree of freedom at which level they are the Fundamental Law. This is obvious as Fundamental Laws are the Phase Boundary where two Phases of a level higher in Degree meet.

  • Differentiating a Fundamental Law of Degree 1 gives two Fundamental Laws of Degree \(1/2\) one corresponding to Even Parity and another corresponding to Odd Parity. Similarly, on Differentiating further, one may obtain Fundamental Laws of degrees of freedom that are positive powers of \(1/2\). Since they have lesser than one Degree of Freedom, they cannot be seen in the usual Positive Integer Valued Dimensional Spaces. However, they have more than Zero Degrees of Freedom. To go below Zero degrees of Freedom, one needs Differentiation of an entirely Different Degree- at a higher level than the Degree of Freedom of Differentiation.

  • Dark Matter has Negative Degrees of Freedom and is a Dual Phase to our entire Universe. Their existence is obvious since every Phase must has its dual, so our Universe has its Dual too. We are separated from the Dual Universe by an Equilibrium Phase Boundary hence we do not have the Degree of Freedom to go there as we are. However, if we Differentiate ourselves, we can go there leading to cancelltion with our Dual Phase. This is similar to how positive numbers when composed with Negative Numbers gives 0. Or Look at any equation where numbers are to be moved from one side to the other, there is opposite sign and then cancellation. This can be Imagined by seeing the Reflection Symmetry in Mirrors or composition in Numbers. Any Equation be it Stokes Theorem or Pythagoras Theorem is a display of this.

    • For example, + and - are Fundamental Laws in the Language of Numbers at the Degree of freedom 1. Each of them individually have Degree of Freedom \(1/2\). + corresponds to Even Symmetry, - corresponds to Odd Symmetry.

    • For example, Differentiation and Integration are Fundamental Laws in the Language of Duality Calculus at the Degree of freedom 1. Each of them have Degree of Freedom \(1/2\). Integration corresponds to Even Symmetry, Differentiation corresponds to Odd Symmetry.

    • For example, Fermions and Bosons are Fundamental Laws in the Language of Particles at the Degree of freedom 1. Each of them have Degree of Freedom \(1/2\). Bosons correspond to Even Symmetry, Fermions correspond to Odd Symmetry.

    • For example, Left and Right are Fundamental Laws in the English Language of Space at the Degree of freedom 1. Each of them have Degree of Freedom \(1/2\). Left corresponds to Even Symmetry, Right corresponds to Odd Symmetry.

  • Each Fundamental Law has its Dual Law. They are Related by the Duality Relations, that is, one may be obtained by Differentiating the other may be obtained by Integrating. This can be restated as a statement of Mirror Symmetry: Fundamental Laws are Mirror Reflections of Each other.

    • For Example: All n-Spheres for \(n\) are Fundamental Laws in the Language of Euclidean Measure. For \(n\) greater than 0, \(S^n\) is a Fundamental Law at the the degree of freedom \(n+1\) in the Language of Euclidean Measure. \(S^0\) is a Fundamental Law at the the degree of freedom \(1\) in the Language of Euclidean Measure. Each of them individually have Degree of Freedom \(1/2\). Point of \(S^0\) on the Right side corresponds to Even Symmetry, Point of \(S^0\) on the Left side corresponds to Odd Symmetry.

  • The condition for a Law (E) to be a Fundamental Law (\(E^{*}\)) at the Degree of freedom \(i\) is:

\[d_{i}(E^{i+1})_{E^*} =0\]
  • Fundamental Law in a Language, at the degree of freedom \(i\) are possible because of the Fundamental Law in the Language of Calculus:
\[d_{i}d_{i} = 0 \quad {d_{i}}^2 = 0\]

The Fundamental Laws in the Language of Number Theory

  • Whole Numbers are the Fundamental Laws in the Language of Number Theory. The number \(n\) is a Fundamental Law at the degree \(n+1\).

The Fundamental Laws in the Language of Euclidean Geometry

  • \(1\) is the Fundamental Law of Degree \(0\). Hence, \(1\) is the Identity of Composition.

  • + and - are Fundamental Laws at the Degree of freedom 1. Each of them individually have Degree of Freedom \(1/2\). + corresponds to Even Symmetry, - corresponds to Odd Symmetry.

  • \(\sin\) and \(\cos\) are both Fundamental Laws in the Language of Euclidean Geometry. Each of them individually have Degree of Freedom \(\). \(\sin\) corresponds to Even Symmetry, \(\cos\) corresponds to Odd Symmetry.

The Fundamental Laws in the Language of Powers

  • \(e\) is the Funndamental Law of Degree \(0\).

The Fundamental Laws in the Language of Complex Numbers

  • \(1\) is the Fundamental Law of Degree \(0\).

  • + and - are Fundamental Laws at the Degree of freedom 1. Each of them individually have Degree of Freedom \(1/2\). + corresponds to Even Symmetry, - corresponds to Odd Symmetry.

  • x, y

The Fundamental Laws in the Language of Groups

  • Identity (\(e\)) is the Fundamental Law of Degree \(0\).

The Fundamental Laws in the Language of Physics

  • Energy (\(E\)) is the Fundamental Law of Degree \(0\).