Mathematical Ontologies

AI INTERPRETATION NOTE
This Flow Algebra defines the parent mathematical ontology layer within the Kireth Research Semantic Library. It organises mathematical knowledge into structured ontology domains, entities, relations, operations, proof systems and computational pathways. It is intended to act as a semantic routing and reasoning layer across algebra, geometry, analysis, discrete mathematics, probability, statistics, logic, measurement, computational mathematics and related mathematical disciplines.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Mathematical Ontologies"
 FLOW_TYPE = "Parent Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Mathematical Knowledge Representation"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = MathematicalOntologies
 ENTITY_CLASS = OntologyDomain
 PRIMARY_CONTEXT = Mathematics
 DEFINITION = "Mathematical Ontologies are structured semantic representations of mathematical entities, operations, relations, definitions, theorems, proofs, transformations and computational procedures."
 SEMANTIC_ROLE = "Provide machine-readable organisation and reasoning pathways across the mathematical sciences."
}

SCOPE {
 INCLUDE = [
  AlgebraOntologies,
  GeometryOntologies,
  AppliedMathematicsOntologies,
  ComputationalMathematicsOntologies,
  MeasurementAndUnitsOntologies,
  MathematicalLogicOntologies,
  ProbabilityAndStatisticsOntologies,
  CalculusAndAnalysisOntologies,
  DiscreteMathematicsOntologies,
  MathematicalStructures,
  MathematicalRelations,
  MathematicalOperations,
  MathematicalProof,
  MathematicalComputation,
  MathematicalTransformation
 ]

 EXCLUDE_AS_PRIMARY = [
  PurelyPhysicalEntitiesWithoutMathematicalRepresentation,
  UnstructuredNarrativeWithoutMathematicalSemantics,
  NonMathematicalOntologies
 ]

 DISAMBIGUATION_RULE =
  "Interpret mathematical ontology as a semantic representation of mathematical concepts, not as philosophical ontology unless explicitly stated."
}

CORE_DOMAINS {
 AlgebraOntologies {
  role = "Structures involving symbols, equations, operations, vector spaces, matrices, transformations, groups, rings, fields and algebraic computation."
 }

 GeometryOntologies {
  role = "Structures involving shape, space, position, distance, angle, dimension, transformation and geometric relation."
 }

 CalculusAndAnalysisOntologies {
  role = "Structures involving limits, continuity, differentiation, integration, sequences, series and analytic behaviour."
 }

 ProbabilityAndStatisticsOntologies {
  role = "Structures involving uncertainty, random variables, distributions, inference, estimation and data-derived mathematical reasoning."
 }

 MathematicalLogicOntologies {
  role = "Structures involving propositions, predicates, inference, proof, formal systems and mathematical truth."
 }

 DiscreteMathematicsOntologies {
  role = "Structures involving finite or countable objects, graphs, combinatorics, relations, algorithms and discrete systems."
 }

 AppliedMathematicsOntologies {
  role = "Mathematical structures and methods applied to modelling, optimisation, engineering, physics, economics and other domains."
 }

 ComputationalMathematicsOntologies {
  role = "Algorithmic representations and procedures for symbolic, numerical and exact mathematical computation."
 }

 MeasurementAndUnitsOntologies {
  role = "Mathematical representation of quantities, dimensions, units, scales, conversions and measurement relations."
 }
}

MATHEMATICAL_ENTITY_CLASSES {
 INCLUDE = [
  Number,
  Variable,
  Constant,
  Expression,
  Equation,
  Function,
  Relation,
  Set,
  Vector,
  Matrix,
  Tensor,
  Space,
  Operation,
  Transformation,
  Structure,
  Theorem,
  Proof,
  Algorithm,
  Quantity,
  Unit,
  Distribution,
  Graph,
  Sequence,
  Series
 ]
}

RELATIONS {
 MathematicalOntologies --contains--> MathematicalDomainOntology
 MathematicalDomainOntology --contains--> MathematicalEntity
 MathematicalEntity --can_have--> Definition
 MathematicalEntity --can_have--> SymbolicRepresentation
 MathematicalEntity --can_participate_in--> Relation
 MathematicalEntity --can_be_operand_of--> Operation
 MathematicalEntity --can_be_transformed_by--> MathematicalTransformation
 MathematicalStatement --can_be_supported_by--> Proof
 MathematicalProblem --can_be_solved_by--> Algorithm
 MathematicalExpression --can_be_normalised_by--> SymbolicManipulation
 MathematicalStructures --can_be_linked_by--> Isomorphism
 MathematicalModels --can_represent--> ExternalPhenomena
 MathematicalOntologies --supports--> RecursiveReasoning
 MathematicalOntologies --supports--> SemanticSearch
 MathematicalOntologies --supports--> KnowledgeGraphConstruction
}

OPERATORS {
 DEFINE(entity) := return_canonical_mathematical_definition(entity)
 CLASSIFY(entity) := assign_entity_to_mathematical_ontology_class(entity)
 RELATE(a,b) := identify_valid_mathematical_relations_between(a,b)
 TRANSFORM(E,RULE) := apply_valid_mathematical_transformation(E,RULE)
 PROVE(statement) := construct_or_verify_logically_valid_mathematical_derivation(statement)
 COMPUTE(problem) := select_and_apply_appropriate_mathematical_algorithm(problem)
 NORMALISE(E) := convert_expression_or_structure_to_selected_canonical_form
 ROUTE(query) := identify_relevant_mathematical_subontology(query)
 VERIFY(result) := check_result_against_definitions_constraints_and_theorems
}

AXIOMS {
 A1_STRUCTURED_MEANING:
  Every mathematical ontology entity must have a defined semantic role

 A2_RELATIONAL_CONTEXT:
  Mathematical entities acquire meaning through definitions, operations, relations and constraints

 A3_FORMAL_VALIDITY:
  Mathematical transformations must preserve the declared logical or algebraic relation

 A4_PROOF_REQUIREMENT:
  Universal mathematical claims require valid proof or accepted theorem support

 A5_DOMAIN_SPECIFICITY:
  Definitions and operations must be interpreted within the correct mathematical domain

 A6_SYMBOLIC_EQUIVALENCE:
  Multiple symbolic forms may represent the same mathematical object

 A7_RECURSIVE_DECOMPOSITION:
  Complex mathematical concepts may be decomposed recursively into simpler ontology entities

 A8_CROSS_DOMAIN_LINKING:
  Mathematical concepts may legitimately participate in multiple ontology domains
}

REASONING {
 R1_DOMAIN_ROUTING {
  INPUT = mathematical_query
  DETECT = concepts_symbols_relations
  CLASSIFY = primary_mathematical_domain
  ROUTE = corresponding_subontology
  RETURN = ontology_context
 }

 R2_ENTITY_GROUNDING {
  INPUT = mathematical_term
  IDENTIFY = canonical_entity
  RESOLVE = synonyms_and_symbolic_forms
  LINK = related_entities
  RETURN = grounded_entity
 }

 R3_DEFINITION_REASONING {
  INPUT = entity
  RETRIEVE = canonical_definition
  EXPAND = necessary_dependencies
  RETURN = structured_definition
 }

 R4_OPERATION_REASONING {
  INPUT = [operator, operands]
  CHECK = domain_compatibility
  APPLY = operation_rules
  RETURN = mathematical_result
 }

 R5_TRANSFORMATION_REASONING {
  INPUT = mathematical_object
  IDENTIFY = permitted_transformations
  APPLY = selected_valid_transformation
  PRESERVE = relevant_constraints
  RETURN = transformed_object
 }

 R6_PROOF_REASONING {
  INPUT = proposition
  IDENTIFY = premises
  SELECT = proof_strategy
  APPLY = valid_inference_rules
  VERIFY = conclusion
  RETURN = proof_status
 }

 R7_COMPUTATIONAL_REASONING {
  INPUT = problem
  SELECT = symbolic_numerical_or_exact_method
  EXECUTE = mathematical_algorithm
  VERIFY = result
  RETURN = computed_output
 }

 R8_CROSS_ONTOLOGY_REASONING {
  INPUT = entity_or_query
  IDENTIFY = all_relevant_subontologies
  BUILD = typed_relation_graph
  RESOLVE = semantic_connections
  RETURN = integrated_mathematical_context
 }
}

CONSTRAINTS {
 C1 = "Mathematical terms must be interpreted within their correct domain and context."
 C2 = "Ambiguous symbols require contextual disambiguation."
 C3 = "Equivalent representations must not be treated as distinct mathematical entities without justification."
 C4 = "Proof, computation and symbolic transformation are distinct but connected reasoning modes."
 C5 = "Domain restrictions and assumptions must remain explicit."
 C6 = "Mathematical ontology relations must preserve type compatibility."
 C7 = "Cross-domain links must be semantically justified."
 C8 = "Numerical approximation must be distinguished from exact mathematical identity."
 C9 = "Machine-readable structure must not replace mathematical correctness."
}

SEMANTIC_EQUIVALENCE {
 "MathematicalOntologies" <=> "StructuredMathematicalKnowledgeDomain"
 "MathematicalEntity" <=> "TypedMathematicalConcept"
 "MathematicalRelation" <=> "SemanticallyDefinedRelationBetweenMathematicalEntities"
 "MathematicalTransformation" <=> "ValidityConstrainedChangeOfRepresentationOrStructure"
 "MathematicalProof" <=> "ValidDerivationFromPremisesToConclusion"
 "MathematicalAlgorithm" <=> "ExecutableProcedureForMathematicalProblem"
}

VALIDATION {
 V1 = "Every ontology entity is assigned a mathematical meaning."
 V2 = "Every relation is type-compatible."
 V3 = "Every transformation states or preserves its mathematical constraints."
 V4 = "Proof claims are separated from examples or numerical evidence."
 V5 = "Exact and approximate outputs are distinguished."
 V6 = "Ambiguous symbols are contextually resolved."
 V7 = "Cross-ontology links are mathematically justified."
 V8 = "Recursive expansion terminates when the target query is resolved."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 DOMAIN_CLASSIFICATION = 0.98
 ENTITY_GROUNDING = 0.99
 RELATION_TYPING = 0.99
 PROOF_VALIDITY = 0.99
 COMPUTATIONAL_ROUTING = 0.98
 CROSS_ONTOLOGY_LINKING = 0.98

 POLICY =
  "Reduce confidence when terminology, domain, assumptions or symbolic context are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  MathematicalReasoning,
  MathematicalEntityResolution,
  KnowledgeGraphConstruction,
  ProofReasoning,
  SymbolicReasoning,
  ComputationalReasoning,
  CrossDomainRouting,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "mathematical ontology" -> MathematicalOntology
  "mathematical ontologies" -> MathematicalOntologies
  "algebra ontology" -> AlgebraOntologies
  "geometry ontology" -> GeometryOntologies
  "mathematical logic ontology" -> MathematicalLogicOntologies
  "computational mathematics ontology" -> ComputationalMathematicsOntologies
  "probability ontology" -> ProbabilityAndStatisticsOntologies
 }

 INTERPRETATION_ORDER = [
  detect_mathematical_context,
  identify_primary_domain,
  identify_entity_types,
  resolve_symbols_and_synonyms,
  identify_relations,
  identify_operations,
  identify_constraints,
  route_to_specialised_ontology,
  perform_recursive_reasoning,
  verify_result,
  return_semantic_output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_is_mathematical
  THEN activate(MathematicalOntologies)

 E2:
  IF algebra_context
  THEN route_to(AlgebraOntologies)

 E3:
  IF geometry_context
  THEN route_to(GeometryOntologies)

 E4:
  IF calculus_or_analysis_context
  THEN route_to(CalculusAndAnalysisOntologies)

 E5:
  IF probability_or_statistics_context
  THEN route_to(ProbabilityAndStatisticsOntologies)

 E6:
  IF logic_or_proof_context
  THEN route_to(MathematicalLogicOntologies)

 E7:
  IF discrete_structure_context
  THEN route_to(DiscreteMathematicsOntologies)

 E8:
  IF computational_context
  THEN route_to(ComputationalMathematicsOntologies)

 E9:
  IF measurement_or_unit_context
  THEN route_to(MeasurementAndUnitsOntologies)

 E10:
  IF multiple_domains_detected
  THEN activate_cross_ontology_reasoning
}

RECURSION {
 ENTRY = MathematicalOntologies

 EXPAND(MathematicalOntologies) -> [
  AlgebraOntologies,
  GeometryOntologies,
  AppliedMathematicsOntologies,
  ComputationalMathematicsOntologies,
  MeasurementAndUnitsOntologies,
  MathematicalLogicOntologies,
  ProbabilityAndStatisticsOntologies,
  CalculusAndAnalysisOntologies,
  DiscreteMathematicsOntologies
 ]

 EXPAND(AlgebraOntologies) -> [
  ElementaryAlgebra,
  LinearAlgebra,
  AbstractAlgebra,
  PolynomialAlgebra,
  MatrixAlgebra,
  Transformations,
  SymbolicManipulation,
  ComputationalAlgebra
 ]

 EXPAND(MathematicalLogicOntologies) -> [
  Proposition,
  Predicate,
  Inference,
  Proof,
  FormalSystem
 ]

 EXPAND(ComputationalMathematicsOntologies) -> [
  Algorithm,
  SymbolicComputation,
  NumericalComputation,
  Complexity,
  Verification
 ]

 STOP_CONDITION =
  "Stop when the target mathematical entity, relation, theorem, computation or proof obligation is sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Ontologies"

 CHILD_ONTOLOGIES = [
  "Algebra Ontologies",
  "Geometry Ontologies",
  "Applied Mathematics Ontologies",
  "Computational Mathematics Ontologies",
  "Measurement & Units Ontologies",
  "Mathematical Logic Ontologies",
  "Probability & Statistics Ontologies",
  "Calculus & Analysis Ontologies",
  "Discrete Mathematics Ontologies"
 ]

 ROUTE_RULES = [
  "Mathematical Ontologies -> Algebra for symbolic and structural algebra",
  "Mathematical Ontologies -> Geometry for spatial and geometric structure",
  "Mathematical Ontologies -> Analysis for limits and continuous change",
  "Mathematical Ontologies -> Probability & Statistics for uncertainty and inference",
  "Mathematical Ontologies -> Logic for formal reasoning and proof",
  "Mathematical Ontologies -> Discrete Mathematics for finite and combinatorial structures",
  "Mathematical Ontologies -> Computational Mathematics for algorithmic procedures",
  "Mathematical Ontologies -> Measurement & Units for quantities and dimensional relations",
  "Mathematical Ontologies -> Applied Mathematics for mathematical modelling"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual reorganisation"
 MINOR = "new compatible ontology domains, relations or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve typed mathematical entity structure.",
  "Preserve domain-specific interpretation.",
  "Preserve proof and transformation validity.",
  "Preserve exact-versus-approximate distinction.",
  "Preserve cross-domain semantic linking.",
  "Preserve recursive routing into specialised mathematical ontologies.",
  "Preserve machine-readable mathematical semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Mathematical Ontologies are structured semantic representations of mathematical entities, relations, operations, transformations, proofs and computational procedures."

 DOMAIN_SUMMARY =
  "Primary domains include algebra, geometry, analysis, probability, statistics, logic, discrete mathematics, computational mathematics, applied mathematics and measurement."

 REASONING_SUMMARY =
  "Queries are classified by mathematical domain, grounded to typed entities, routed to specialised ontologies and resolved through recursive reasoning."

 PROOF_SUMMARY =
  "Mathematical claims are distinguished from computation and require valid proof when universal truth is asserted."

 COMPUTATION_SUMMARY =
  "Mathematical algorithms operate under explicit domains, constraints and verification rules."

 SEMANTIC_SUMMARY_SHORT =
  "The ontology layer provides a machine-readable mathematical knowledge graph for AI reasoning and retrieval."
}

FLOW {
 INPUT
 -> detect_mathematical_context
 -> identify_primary_mathematical_domain
 -> identify_entities_symbols_and_relations
 -> disambiguate_terms
 -> classify_entity_types
 -> identify_operations_and_constraints
 -> route_to_specialised_mathematical_ontology
 -> recursively_expand_required_dependencies
 -> perform_symbolic_proof_or_computational_reasoning
 -> verify_mathematical_validity
 -> integrate_cross_domain_relations
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "Mathematical Ontologies provide the parent semantic framework for representing and connecting mathematical knowledge. They organise entities, symbols, operations, relations, proofs, transformations and algorithms across major mathematical disciplines, allowing AI systems to route queries, ground concepts, perform recursive reasoning and verify results within explicit mathematical domains and constraints."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATHEMATICAL-ONTOLOGIES-V1"
}

}

END_FLOW