Principia Mathematica tried to construct the whole mathematics using calculus of proposition as its basic tool. The calculus itself is based on five simple axioms: tautology, addition, permutation, Association ad Summation.
Based on this calculus, the whole Boolean algebra can be constructed. However, Boolean algebra can also be built on the base three axioms of Sheffer
which used only one logical NAND operation.In the appendix of his book Laws of Form, George Spencer-Brown has proven that the three Sheffer axioms can be proven as consequences of his two simple axioms: position and transposition.
This means that the five axioms of the Principia's calculus of proposition can be derived from the two axioms of brownian algebra that can be simulated by the game of things as it is shown in my last blog.
In this blog we will represent the Principia axioms into a configuration of boxes and colored balls and challenge you to transform them to a single box representing TRUE. If you can do it, then you already proved they are the consequences of Brownian algebra.
My proof is using the following first three consequences of Brownian algebra considered as rules of transformations in the Brownian game of things
Tautology can be proved by using generation twice ended by doing integration.
Addition, permutation and association can be proved in a similar manner.
Summation can be proved firstly by removing the red ball within the box, then removing the contents of the first box using generation ended by doing integration.
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