Introduction to Logic Games of Thing

Armahedi Mahzar (c) 2013

When I was retired as a lecturer of physics 14 years ago in Indonesia, I intended to become a student again. This time the program of study is mathematics that I loved from the beginning of my school days, because in my study as a university student in the theoretical physics I used many types of mathematics that are not used by the engineer.

 

Engineering, as it is known when I was a student in the university, mostly only knew the 2 dimensional complex numbers with an imaginary number besides the real number. In particle physics I was taught the algebra of the Pauli 2×2  matrices, the 3×3 matrices of Gell-Mann  and Dirac  4×4 matrices each of which can be used as as units for 4, 8 and 16 dimensional algebraic numbers. Such diversity of multidimensional numbers amazed me, so I determined to study the general form of it during my retirement.

 

When I entered the cyber-university I joined a cyber-class or a mailing list studying hypernumbers of Charles Muses . In my study of 16 dimensional numbers there, I faced an issue of algebraic logic, so I went to the lawsofform mailing list discussing the Laws of Forms book of George Spencer-Brown . From it I learn to understand Spencer-Brown VOID CROSS symbolism for his primary algebra of Laws of Form.

From my encounter with Laws of Form, I started my journey to explore the beautiful simplicity of the Boolean space as the part of the Platonic space of mathematical forms. The axiomatic simplicity of it is just one side of the journey. In the following blogs I will trace the steps toward the discovery of the astonishing concrete games of thing as the alternative to the abstract algebra of mathematical symbols. Let trace the history of symbolic logic.

 

Logic as 1-Dimensional Algebra of Symbols

Long before the strange looking algebra of Spencer-Brown appears, logic can be formulated as the algebra of symbols as it is invented by George Boole in the 19th century. The Boolean algebra is based on an arithmetic of two values, 1 and 0, which symbolized the logical values TRUE and FALSE respectively. The arithmetic operations TIMES and PLUS is symbolizing the logical operation AND and OR respectively. The NOT operation is symbolized by the complementation 1 MINUS.

With this arithmetic symbolization, the reasoning process is becoming an algebraic calculation or solving algebraic equation. The nineteenth century is flowered by the refinement of the symbolic representation of the logical reasoning. This symbolization can be classified into two kinds of symbols: the algebraic and the linguistic. The algebraic formalization is matured in the Boole -Peirce -Schroder mathematical logic and the linguistic formulation is matured in Frege -Russel  symbolic calculus.

The fact is both Peirce and Frege has developed different 2-dimensional pictorial symbols. Peirce has developed an enclosure system of ovals and Frege has developed a branching system of trees. Unfortunately, Schroder linearized Peircean system and Russel linearized Fregean symbolization.   In the 20th century the Peircean system revived by George Brown into a crossing system  of vinculus as an alternative of the enclosure system of ovals.

Fortunately, William Bricken  had rewritten the crossing system to a bracketing system using two linear symbols of parentheses, as it has been done by Peirce before, so we have a linear symbolization of logic that can easily be typed in the screen of a computer. However, such linear system is not a replacement or reduction of 2-d systems, it is just a projection of it in 1-d  string of symbols. So, permutation of symbols representing classes and propositions is accepted implicitly as the same representation of 2d configuration of symbols.

Transforming the Symbolic Algebra of Logic

From the Brownian Cross Void symbolization of Boolean algebra, I met the Box Algebra of Louis Kauffman  which become the first stepping stone to my journey for simple visualization of logic ending in my discovery of the varieties of concrete games of things for logic.  The steps of discovery is the following links:

 

Step 1: Coloring Box Algebra of Logic
Step 2: Colorful Bar Algebra of Logic
Step 3: Colorful V algebra of Logic
Step 4: Colorful T algebra of Logic
Step 5: Colorful Stick algebra of Logic
Step 6: Wonderful Card Algebra of Logic
Step 7: Variety of Logic Games of Thing