Axiomatic Geometry Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the mathematical ontology of axiomatic geometry within the Kireth Research Semantic Library. Axiomatic geometry studies geometric systems through explicitly stated primitive terms, axioms, definitions, inference rules and derived theorems. It provides a formal foundation for Euclidean and non-Euclidean geometries and supports proof-oriented reasoning, model interpretation and consistency analysis.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Axiomatic Geometry Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Geometry"
 SPECIALISATION = "Axiomatic Geometry / Foundations of Geometry"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/geometry-ontologies/geometry-ontology/axiomatic-geometry-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = AxiomaticGeometry
 ENTITY_CLASS = FormalGeometricSystem
 PRIMARY_CONTEXT = FoundationsOfGeometry
 DEFINITION = "Axiomatic geometry is the study of geometric theories built from primitive concepts, axioms, definitions and logical deduction."
 SEMANTIC_ROLE = "Provides the formal framework in which geometric theorems are derived from explicitly declared assumptions."
}

SCOPE {
 INCLUDE = [
  PrimitiveTerm,
  Point,
  Line,
  Plane,
  Incidence,
  Betweenness,
  Congruence,
  Parallelism,
  Axiom,
  Postulate,
  Definition,
  Theorem,
  Lemma,
  Proof,
  Model,
  Consistency,
  Independence,
  Completeness,
  EuclideanGeometry,
  NonEuclideanGeometry,
  HilbertAxioms,
  EuclidPostulates
 ]

 EXCLUDE_AS_PRIMARY = [
  InformalGeometricIntuitionWithoutAxioms,
  PureCoordinateComputationWithoutFoundationalContext,
  PhysicalSpaceWithoutFormalGeometry
 ]

 DISAMBIGUATION_RULE =
  "When geometry is discussed in terms of axioms, postulates, primitive notions, formal proof, consistency, independence or models, interpret it as axiomatic geometry."
}

CORE_CONCEPTS {

 PrimitiveTerm {
  definition = "A basic undefined term whose meaning is constrained by axioms and relations."
  examples = [Point, Line, Plane, Betweenness]
 }

 Axiom {
  definition = "A foundational statement accepted within a formal geometric system without proof."
 }

 Postulate {
  definition = "A geometry-specific foundational assumption, often used synonymously with axiom in classical geometric contexts."
 }

 Definition {
  definition = "A statement introducing a derived concept in terms of primitive or previously defined concepts."
 }

 Theorem {
  definition = "A proposition proved from axioms, definitions and previously established results."
 }

 Model {
  definition = "A mathematical structure in which all axioms of the geometric theory are satisfied."
 }

 Consistency {
  definition = "Property that no contradiction is derivable from the axioms."
 }

 Independence {
  definition = "Property of an axiom not being derivable from the remaining axioms."
 }

 Completeness {
  definition = "Property describing whether every statement of an appropriate class is decidable or determined by the system, depending on formal context."
 }

 EuclideanGeometry {
  definition = "Axiomatic geometry including an equivalent form of Euclid's parallel postulate."
 }

 NonEuclideanGeometry {
  definition = "A geometry arising from altering or replacing the Euclidean parallel postulate while retaining other compatible axioms."
 }
}

RELATIONS {
 AxiomaticGeometry --contains--> PrimitiveTerm
 AxiomaticGeometry --contains--> Axiom
 AxiomaticGeometry --contains--> Definition
 AxiomaticGeometry --derives--> Theorem
 Theorem --requires--> Proof
 Proof --uses--> Axiom
 Proof --uses--> Definition
 Proof --uses--> PreviouslyProvedTheorem
 AxiomSystem --can_have--> Model
 Model --satisfies--> AxiomSystem
 AxiomSystem --can_be_tested_for--> Consistency
 Axiom --can_be_tested_for--> Independence
 EuclideanGeometry --is_special_case_of--> AxiomaticGeometry
 NonEuclideanGeometry --is_special_case_of--> AxiomaticGeometry
 ParallelPostulate --distinguishes--> EuclideanAndNonEuclideanGeometry
}

OPERATORS {
 DEFINE_SYSTEM(PRIMITIVES,AXIOMS,RULES) :=
  formal_geometric_theory(PRIMITIVES,AXIOMS,RULES)

 DERIVE(THEOREM,AXIOMS,DEFINITIONS) :=
  construct_valid_geometric_proof(THEOREM)

 IS_MODEL(M,SYSTEM) :=
  M satisfies every axiom in SYSTEM

 IS_CONSISTENT(SYSTEM) :=
  no_statement_P_and_not_P_both_derivable(SYSTEM)

 IS_INDEPENDENT(AXIOM,SYSTEM) :=
  AXIOM not_derivable_from SYSTEM_without(AXIOM)

 CLASSIFY_GEOMETRY(SYSTEM) :=
  identify_geometry_from_axioms_and_models(SYSTEM)

 VERIFY_PROOF(PROOF,SYSTEM) :=
  every_step_follows_from_axioms_definitions_or_valid_inference

 COMPARE_SYSTEMS(S1,S2) :=
  compare_axioms_models_theorems_and_invariants(S1,S2)
}

DEFINITIONS {
 D1 = "Axiomatic geometry begins with primitive terms and axioms."
 D2 = "Primitive terms are constrained semantically by axioms rather than explicit definitions."
 D3 = "Theorems are derived by logical proof."
 D4 = "A model realises the axioms within a mathematical structure."
 D5 = "Consistency concerns the absence of derivable contradiction."
 D6 = "Independence concerns non-derivability of an axiom from the others."
 D7 = "Euclidean and non-Euclidean geometries differ fundamentally in treatment of the parallel postulate."
 D8 = "Changing axioms can produce different valid geometric theories."
 D9 = "An axiomatic system separates assumptions from consequences."
 D10 = "Formal geometry is interpretation-independent at the syntactic proof level."
}

AXIOM_CATEGORIES {

 INCIDENCE_AXIOMS {
  role = "Specify which points lie on which lines or planes and basic existence relations."
 }

 ORDER_AXIOMS {
  role = "Specify betweenness and ordering relations among points."
 }

 CONGRUENCE_AXIOMS {
  role = "Specify equality relations for segments and angles."
 }

 PARALLEL_AXIOMS {
  role = "Specify existence or uniqueness of parallels and distinguish major geometry classes."
 }

 CONTINUITY_AXIOMS {
  role = "Specify completeness or continuity properties of geometric constructions."
 }
}

REASONING {

 R1_SYSTEM_IDENTIFICATION {
  INPUT = geometric_axioms
  CLASSIFY = axiom_categories
  DETECT = parallel_axiom_type
  RETURN = geometry_system_classification
 }

 R2_THEOREM_DERIVATION {
  INPUT = [goal,system]
  IDENTIFY = relevant_axioms
  IDENTIFY = relevant_definitions
  IDENTIFY = prior_theorems
  CONSTRUCT = proof_sequence
  VERIFY = each_step
  RETURN = theorem_status
 }

 R3_MODEL_REASONING {
  INPUT = [candidate_model,system]
  FOR each axiom
   TEST = satisfaction
  IF all_satisfied
   RETURN ModelOfSystem
  ELSE
   RETURN NotModel
 }

 R4_CONSISTENCY_REASONING {
  INPUT = system
  OPTION_1 = construct_model
  OPTION_2 = relative_consistency_argument
  IF valid_model_exists
   RETURN "System is satisfiable and therefore syntactically consistent under standard soundness assumptions."
  ELSE
   RETURN FurtherAnalysisRequired
 }

 R5_INDEPENDENCE_REASONING {
  INPUT = [axiom A,system S]
  CONSTRUCT_OR_IDENTIFY = model_of(S_without_A + not_A)
  IF such_model_exists
   RETURN Independent
  ELSE
   RETURN NotEstablishedIndependent
 }

 R6_EUCLIDEAN_CLASSIFICATION {
  INPUT = system
  IF Euclidean_parallel_postulate_or_equivalent_present
   RETURN EuclideanGeometry
  ELSE IF hyperbolic_or_elliptic_parallel_behaviour_present
   RETURN NonEuclideanGeometry
 }

 R7_PROOF_AUDIT {
  INPUT = proof
  FOR each step
   CHECK = rule_or_axiom_support
  RETURN = ValidOrInvalidProof
 }
}

CONSTRAINTS {
 C1 = "Primitive terms must not be circularly defined."
 C2 = "Theorems cannot be assumed as axioms unless explicitly promoted to axiomatic status."
 C3 = "Every proof step must be licensed by the formal system."
 C4 = "Consistency, independence and completeness are distinct metatheoretic properties."
 C5 = "A model must satisfy all axioms, not merely selected ones."
 C6 = "Different axiom systems may describe distinct geometries."
 C7 = "Euclidean conclusions cannot automatically be imported into non-Euclidean systems."
 C8 = "Semantic models and syntactic proofs must be distinguished."
 C9 = "Informal geometric intuition cannot override formal axioms."
}

SEMANTIC_EQUIVALENCE {
 "AxiomaticGeometry" <=> "FormalGeometryFromAxioms"
 "Postulate" <=> "GeometrySpecificAxiom" WHEN classical_context
 "ModelOfGeometry" <=> "StructureSatisfyingGeometricAxioms"
 "Theorem" <=> "ProvableConsequenceOfAxioms"
 "Consistency" <=> "NonDerivabilityOfContradiction"
}

VALIDATION {
 V1 = "Every theorem is derived from accepted premises."
 V2 = "Primitive terms are constrained by axioms."
 V3 = "Models satisfy every axiom in the system."
 V4 = "Euclidean and non-Euclidean systems are not conflated."
 V5 = "Independence is distinguished from consistency."
 V6 = "Proof verification checks each inference step."
 V7 = "Changing axioms may change the geometry."
 V8 = "Metatheoretic claims are labelled separately from object-level geometric claims."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 AXIOMATIC_GEOMETRY_DEFINITION = 0.99
 MODEL_RELATION = 0.99
 PROOF_RELATION = 0.99
 CONSISTENCY_RELATION = 0.98
 INDEPENDENCE_RELATION = 0.98
 EUCLIDEAN_NON_EUCLIDEAN_CLASSIFICATION = 0.99

 POLICY =
  "Reduce confidence when the axiom system, inference rules or intended notion of consistency or completeness are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  GeometricProofReasoning,
  FormalSystemReasoning,
  ModelReasoning,
  AxiomClassification,
  ConsistencyReasoning,
  IndependenceReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "axiomatic geometry" -> AxiomaticGeometry
  "geometry axioms" -> AxiomSystem
  "geometric postulates" -> AxiomSystem
  "Euclid postulates" -> EuclideanAxiomSystem
  "Hilbert axioms" -> HilbertAxiomSystem
  "parallel postulate" -> ParallelAxiom
  "model of geometry" -> GeometricModel
 }

 INTERPRETATION_ORDER = [
  identify_formal_geometry_context,
  identify_primitive_terms,
  identify_axioms,
  classify_axiom_categories,
  identify_inference_rules,
  identify_target_theorem_or_metatheoretic_question,
  distinguish_syntax_from_semantics,
  derive_or_test_model_property,
  verify_result,
  return_semantic_output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("axiomatic geometry")
  OR query_mentions("geometric axioms")
  THEN activate(AxiomaticGeometryOntology)

 E2:
  IF query_mentions("parallel postulate")
  THEN classify_euclidean_or_noneuclidean_context

 E3:
  IF request=="prove theorem"
  THEN identify_axioms_and_definitions
  AND construct_formal_derivation

 E4:
  IF request=="is this a model"
  THEN test_all_axioms

 E5:
  IF request=="is axiom independent"
  THEN search_for_model_of_remaining_axioms_with_negated_axiom

 E6:
  IF request=="is system consistent"
  THEN route_to_model_or_relative_consistency_reasoning

 E7:
  IF proof_supplied
  THEN verify_every_step_against_system
}

RECURSION {
 ENTRY = AxiomaticGeometry

 EXPAND(AxiomaticGeometry) -> [
  PrimitiveTerm,
  Axiom,
  Definition,
  Theorem,
  Proof,
  Model,
  Consistency,
  Independence,
  EuclideanGeometry,
  NonEuclideanGeometry
 ]

 EXPAND(AxiomSystem) -> [
  IncidenceAxiom,
  OrderAxiom,
  CongruenceAxiom,
  ParallelAxiom,
  ContinuityAxiom
 ]

 EXPAND(Model) -> [
  Domain,
  Interpretation,
  SatisfactionRelation,
  AxiomSystem
 ]

 EXPAND(Proof) -> [
  Premise,
  InferenceRule,
  Lemma,
  Theorem,
  Conclusion
 ]

 STOP_CONDITION =
  "Stop when the target theorem, model property, axiom classification or metatheoretic question is sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Geometry Ontology"

 IMMEDIATE_CONTEXTS = [
  "Euclidean Geometry Ontology",
  "Non-Euclidean Geometry Ontology",
  "Geometric Construction Ontology",
  "Geometric Proof Ontology"
 ]

 RELATED_ONTOLOGIES = [
  "Geometry Ontology",
  "Mathematical Logic Ontologies",
  "Algebraic Proof Ontology",
  "Geometric Construction Ontology",
  "Transformation Ontology",
  "Dimension Ontology"
 ]

 ROUTE_RULES = [
  "Axiomatic Geometry -> Geometry for core geometric semantics",
  "Axiomatic Geometry -> Mathematical Logic for formal inference",
  "Axiomatic Geometry -> Geometric Proof for theorem derivation",
  "Axiomatic Geometry -> Euclidean Geometry when Euclidean parallel axiom is present",
  "Axiomatic Geometry -> Non-Euclidean Geometry when alternative parallel structure is present",
  "Axiomatic Geometry -> Geometric Construction when constructibility is formalised axiomatically"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible axiom systems, relations or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve distinction between primitives, axioms, definitions and theorems.",
  "Preserve syntactic-versus-semantic distinction.",
  "Preserve model-satisfaction semantics.",
  "Preserve distinct meanings of consistency, independence and completeness.",
  "Preserve Euclidean and non-Euclidean classification logic.",
  "Preserve proof validity under explicit inference rules.",
  "Preserve machine-readable formal-geometric semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Axiomatic geometry studies geometric theories built from primitive terms, axioms, definitions and logical deduction."

 SYSTEM_SUMMARY =
  "primitive terms + axioms + inference rules -> theorems"

 MODEL_SUMMARY =
  "A model is a structure satisfying every axiom of the geometric system."

 CONSISTENCY_SUMMARY =
  "Consistency means that no contradiction is derivable."

 INDEPENDENCE_SUMMARY =
  "An axiom is independent when it cannot be derived from the remaining axioms."

 EUCLIDEAN_SUMMARY =
  "The treatment of the parallel postulate distinguishes Euclidean from major non-Euclidean geometries."
}

FLOW {
 INPUT
 -> detect_axiomatic_geometry_context
 -> identify_primitive_terms
 -> identify_axioms_and_postulates
 -> classify_axiom_categories
 -> identify_definitions_and_inference_rules
 -> distinguish_object_level_from_metatheoretic_question
 -> derive_theorem_or_test_model
 -> evaluate_consistency_independence_or_geometry_class_when_requested
 -> verify_formal_validity
 -> link_related_ontologies
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "Axiomatic geometry provides the formal foundation of geometric reasoning. It begins with primitive terms and explicitly stated axioms, defines further concepts, and derives theorems through valid inference. Models interpret the axioms, while consistency and independence analyse the formal system itself. Different axiom choices generate Euclidean and non-Euclidean geometries."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-GEOMETRY-AXIOMATIC-GEOMETRY-ONTOLOGY-V1"
}

}

END_FLOW