Neutrosophy as a branch of philosophy has many applications in almost every field of science. It studies neutralities, which are found in almost every real-life problem. This chapter is a review of reproduction of the evolution from the foundation of paradoxism (1980s), then of neutrosophy (1995) and its derivatives (neutrosophic set/logic/probability/statistics), then neutrosophic algebraic structures (2003), and ending with the NeutroAlgebra and AntiAlgebra (2019) as well as the NeutroGeometry and AntiGeometry (2021) of the previous work.
TopFrom Paradoxism To Neutrosophy
Paradoxism is an international movement in science and culture, founded by Florentin Smarandache in 1980s, based on excessive use of antitheses, oxymoron, contradictions, and paradoxes. During three decades (1980-2020) hundreds of authors from tens of countries around the globe contributed papers to 15 international paradoxist anthologies.
In 1995, he extended the paradoxism (based on opposites) to a new branch of philosophy called neutrosophy (based on opposites and their neutral), that gave birth to many scientific branches, such as: neutrosophic logic, neutrosophic set, neutrosophic probability and statistics, neutrosophic algebraic structures, and so on with multiple applications in engineering, computer science, administrative work, medical research, etc.
Neutrosophy is an extension of Dialectics that have derived from the Yin-Yan Ancient Chinese Philosophy.
The Neutrosophic Algebraic Structures are based on neutrosophic numbers of the form a+bI, where a, b are real or complex, while I = literal indeterminacy with I2= I.
TopOperation, Neutrooperation, Antioperation
When we define an operation on a given set, it does not automatically mean that the operation is well-defined. There are three possibilities:
- 1.
The operation is well-defined (also called inner-defined) for all set's elements [degree of truth T = 1] (as in classical algebraic structures; this is a classical Operation). Neutrosophically we write: Operation(1,0,0).
- 2.
The operation if well-defined for some elements [degree of truth T], indeterminate for other elements [degree of indeterminacy I], and outer-defined for the other elements [degree of falsehood F], where (T,I,F) is different from (1,0,0) and from (0,0,1) (this is a NeutroOperation). Neutrosophically we write: NeutroOperation(T,I,F).
- 3.
The operation is outer-defined for all set's elements [degree of falsehood F = 1] (this is an AntiOperation). Neutrosophically we write: AntiOperation(0,0,1).