Armahedi Mahzar (c) 2018
When I finally understood primary algebra in the book entitled Laws of Form
, I thought the calculus of
indication is a notational and axiomatic simplification of the calculus
of proposition as it is formulated by Russel and Whitehead in their book
Principia Mathematica.However, later, I realized that the primary algebra is essentially a pure algebra, not a logical algebra, so it is not really a notational simplification of Boolean algebra. It uses only purely algebraic inference rules: the substitution and replacement rule.
The Primary Algebra only becomes a Boolean algebra if we define AND and IF operations using LoF VOID operation which is interpreted as OR or AND operation. So, augmented Primary Algebra or augmented calculus of indication is isomorphic to the Principia calculus of proposition which used Modus Ponens as its only rule of inference.
Consequently, the augmented calculus of indication which is not using logical inference rule is equivalent to a calculus of proposition using logical rule modus ponens.
For me, it is an amazing fact how the foundation of logic (calculus of proposition) and and the foundation of mathematics (calculus of indication) is actually similar.
Lately, I found out that the primary algebra can be proven, by Louis Kauffman, as a system based on a single axiom, Huntington axiom, using only algebraic rules of inference.
I also found out the Boolean logic has been proven, by Charles Sanders Peirce, as a system based on a single axiom (TRUE symbolized as VOID in Existential Graph System) and five implicational logical rules who are not equal to modus ponens.
Fortunately,
if we replace the single axiom to Consistency (p->p) the five rules
can be reduced to just one rule ( generation (pq)q=(p)q or the third
consequence C3 of the primary algebra. https://issuu.com/armahedimahzar/docs/the_simple_existential_graph_system )
As the consequence, both foundations of logic or mathematics is just a game of symbols with formation rules (definitions and axioms) and transformation rules (equational or implicational rules).
However, when the symbols are represented by things the game of symbols becomes the Game of Things.So the foundation of logic and math is becoming just a game of things.
(The 3D gamification of LoF is discussed in my blog https://integralisme.wordpress.com/2013/06/21/528/ . The 2D gamification of LoF is discussed in my book
)
This finding astonished me. How can the abstract logic and math is equivalent to a concrete game? Why? What is the philosophical implication of this astonishing fact?
As the consequence, both foundations of logic or mathematics is just a game of symbols with formation rules (definitions and axioms) and transformation rules (equational or implicational rules).
However, when the symbols are represented by things the game of symbols becomes the Game of Things.So the foundation of logic and math is becoming just a game of things.
(The 3D gamification of LoF is discussed in my blog https://integralisme.wordpress.com/2013/06/21/528/ . The 2D gamification of LoF is discussed in my book
)This finding astonished me. How can the abstract logic and math is equivalent to a concrete game? Why? What is the philosophical implication of this astonishing fact?
