NeutroGeometry & AntiGeometry
as alternatives and generalizations
of Non-Euclidean Geometry
In our real world there are many neutrosophic triplets of the form (<A>, <neutA>, <antiA>), where <A> is some entity, while <antiA> is its opposite, and <neutA> is the neutral (indeterminate) between them.
A classical Geometry structure has all axioms totally (100%) true. A NeutroGeometry structure has some axioms that are only partially true, and no axiom is totally (100%) false. Whereas an AntiGeometry structure has at least one axiom that is totally (100%) false.
Let's examine several neutrosophic triplets.
1. (<Structure>, <NeutroStructure>, <AntiStructure>) in any field of knowledge
A Structure, in any field of knowledge, is composed of: a non-empty space, populated by some elements, and both (the space and all elements) are characterized by some relations among themselves (such as: operations, laws, axioms, properties, functions, theorems, lemmas, consequences, algorithms, charts, hierarchies, equations, inequalities, etc.), and by their attributes (size, weight, color, shape, location, etc.).
[Of course, when analysing a structure, it counts with respect to what space, elements, relations, and attributes we do it.]
(i) If all these categories (space, elements, relations, attributes) are determinate and 100% true, we deal with a classical Structure.
(ii) If there is some indeterminacy data with respect to any of these categories (space, elements, relations, attributes), or some relation that is partially false, and no relation that is totally false, we have a NeutroStructure.
(iii) If at least one relation is 100% false, we have an AntiStructure.
2. (<Relation>, <NeutroRelation>, <AntiRelation>)
(i) A classical Relation is a relation that is true for all elements of the set (degree of truth T = 1). Neutrosophically we write Relation(1,0,0).
(ii) A NeutroRelation is a relation that is true for some of the elements (degree of truth T), indeterminate for other elements (degree of indeterminacy I), and false for the other elements (degree of falsehood F). Neutrosophically we write Relation(T,I,F), where (T,I,F) is different from (1,0,0) and (0,0,1).
(iii) An AntiRelation is a relation that is false for all elements (degree of falsehood F = 1). Neutrosophically we write Relation(0,0,1).
3. (<Attribute>, <NeutroAttribute>, <AntiAttribute>)
(i) A classical Attribute is an attribute that is true for all elements of the set (degree of truth T = 1). Neutrosophically we write Attribute(1,0,0).
(ii) A NeutroAttribute is an attribute that is true for some of the elements (degree of truth T), indeterminate for other elements (degree of indeterminacy I), and false for the other elements (degree of falsehood F). Neutrosophically we write Attribute(T,I,F), where (T,I,F) is different from (1,0,0) and (0,0,1).
(iii) An AntiAttribute is an attribute that is false for all elements (degree of falsehood F = 1). Neutrosophically we write Attribute(0,0,1).
4. (<Axiom>, <NeutroAxiom>, <AntiAxiom>)
When we define an axiom on a given set, it does not
automatically mean that the axiom is true for all set elements. We have three possibilities again:
(i) The axiom is true for all set's elements (totally true) [degree of truth
T = 1] (as in classical algebraic structures; this is a classical Axiom).
Neutrosophically we write: Axiom(1,0,0).
(ii) The axiom if true for some elements [degree of truth T], indeterminate
for other elements [degree of indeterminacy I], and false for other elements
[degree of falsehood F], where (T,I,F) is different from (1,0,0) and from
(0,0,1) (this is NeutroAxiom). Neutrosophically we write
NeutroAxiom(T,I,F).
(iii) The axiom is false for all set's elements [degree of falsehood F =
1](this is AntiAxiom). Neutrosophically we write AntiAxiom(0,0,1).
5. (<Theorem>, <NeutroTheorem>, <AntiTheorem>)
In any science, a classical Theorem, defined on a given space, is a statement that is 100% true (i.e. true for all elements of the space). To prove that a classical theorem is false, it is sufficient to get a single counter-example where the statement is false. Therefore, the classical sciences do not leave room for partial truth of a theorem (or a statement). But, in our real world and in our everyday life, we have many more examples of statements that are only partially true, than statements that are totally true. The NeutroTheorem and AntiTheorem are generalizations and alternatives of the classical Theorem in any science.
Let's consider a theorem, stated on a given set, endowed with some operation(s).
When we construct the theorem on a given set, it does not automatically mean
that the theorem is true for all set’s elements. We have three possibilities
again:
(i) The theorem is true for all set's elements [totally true] (as in
classical algebraic structures; this is a classical Theorem).
Neutrosophically we write: Theorem(1,0,0).
(ii) The theorem if true for some elements [degree of truth T], indeterminate
for other elements [degree of indeterminacy I], and false 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 NeutroTheorem). Neutrosophically we write:
NeutroTheorem(T,I,F).
(iii) The theorem is false for all set's elements (this is an AntiTheorem).
Neutrosophically we write: AntiTheorem(0,0,1).
And similarly for (Lemma, NeutroLemma, AntiLemma), (Consequence, NeutroConsequence, AntiConsequence), (Algorithm, NeutroAlgorithm, AntiAlgorithm), (Property, NeutroProperty, AntiProperty), etc.
6. (<Geometry>, <NeutroGeometry>, <AntiGeometry>)
(i) A geometric structure who’s all axioms (and theorems,
propositions, etc.) are totally true, and there is no indeterminate data, is called a classical Geometric Structure (or Geometry).
(ii) A geometric structure that has at least one NeutroAxiom (and no AntiAxiom),
or some indeterminate data, is called a NeutroGeometric
Structure (or NeutroGeometry).
(iii) A geometric structure that has at least one AntiAxiom is called an
AntiGeometric Structure (or AntiGeometry).
Therefore, a neutrosophic triplet is formed: (<Geometry>,
<NeutroGeometry>,
<AntiGeometry>),
where <Geometry> can be any classical Euclidean, Projective, Affine,
Differential, Discrete etc. geometric structure.
7. Real Examples of NeutroGeometry and AntiGeometry
7.1. The NeutroGeometry is a generalization of the Hybrid Geometries (SG)
The Hybrid (or Smarandache) Geometries (SG), for the case when an axiom is partially validated and
partially invalidated, and as such it is partially Euclidean and partially Non-Euclidean in the same space).
Because the real geometric spaces are not pure, but hybrid, and the real rules do not uniformly apply to
all space elements, but they have degrees of diversity – applying to some geometrical concepts (point,
line, plane, surface, etc.) in a smaller or bigger degree.
From Prof. Dr. Linfan Mao’s arXiv.org paper Pseudo-Manifold Geometries with Applications,
Cornell University, New York City, USA, 2006, https://arxiv.org/abs/math/0610307: “A
Smarandache geometry is a geometry which has at least one Smarandachely denied axiom
(1969), i.e., an axiom behaves in at least two different ways within the same space, i.e.,
validated and invalided, or only invalided but in multiple distinct ways.”
7.2. The AntiGeometry is a generalization of the Non-Euclidean Geometries
While the Non-Euclidean Geometries [hyperbolic and elliptic geometries] resulted from the total
negation of only one specific axiom (Euclid’s Fifth Postulate), the AntiGeometry results from the total
negation of any axiom [and in general: theorem, concept, idea etc.] and even of more axioms [theorem,
concept, idea, etc.] and in general from any geometric axiomatic system (Euclid’s five postulates,
Hilbert’s 20 axioms, etc.), and the NeutroAxiom results from the partial negation of one or more axioms
[and no total negation of no axiom] from any geometric axiomatic system [1].
References
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