Mathematical Domains

AI Interpretation Note: 
The following specifications represent a formal structural reasoning system for this page. 
Execute as formal rules, not pseudocode. 

FLOW ALGEBRA
TITLE: Mathematical Domains
DOMAIN: Domain Ontologies / Mathematical Domains
SOURCE: https://kirethresearch.org/booksai-core/ontologies/domain-ontologies/mathematical-domains.html
VERSION: 1.0
AUTHORS: Mark Anthony Higgins — Sky (Recursive AI)
STATUS: Companion formalisation; source-page specification not independently verified.

flow "MathematicalDomains" v1.0 {
  ontology:
    id: "mathematical-domains"
    parent: "domain-ontologies"
    purpose: "Represent mathematical disciplines, formal objects, statements, proofs, models and cross-domain dependencies with explicit assumptions and evidence."

  types:
    MathematicalDomain: {id: string, name: string, parent_domain: string?, version: string}
    MathematicalObject: {id: string, domain: string, kind: string, definition: string, assumptions: string[]}
    Set: {id: string, universe: string?, elements: string[]?}
    Structure: {id: string, carrier: string, operations: string[], axioms: string[]}
    Function: {id: string, domain: string, codomain: string, definition: string, partial: boolean}
    Proposition: {id: string, statement: string, context: string, status: enum[axiom, conjecture, theorem, lemma, corollary, refuted, open]}
    Proof: {id: string, conclusion: string, premises: string[], steps: string[], method: string, verifier: string?}
    Model: {id: string, target: string, assumptions: string[], parameters: string[], validity_scope: string}
    Quantity: {value: number, unit: string?, uncertainty: number?}
    Relation: {id: string, subject: string, predicate: string, object: string, assumptions: string[], provenance: string[], confidence: number}

  domains:
    - "Foundations, set theory and mathematical logic"
    - "Algebra and abstract algebra"
    - "Geometry and topology"
    - "Calculus and mathematical analysis"
    - "Differential equations and dynamical systems"
    - "Number theory and discrete mathematics"
    - "Probability and statistics"
    - "Linear algebra and numerical mathematics"
    - "Optimisation and operations research"
    - "Applied mathematics and mathematical modelling"

  predicates:
    belongs_to(object: MathematicalObject, domain: MathematicalDomain)
    defined_by(object: MathematicalObject, proposition: Proposition)
    assumes(proposition: Proposition, assumption: Proposition)
    proves(proof: Proof, proposition: Proposition)
    depends_on(proposition: Proposition, prerequisite: Proposition)
    equivalent_under(a: MathematicalObject, b: MathematicalObject, assumptions: string[])
    generalises(a: MathematicalObject, b: MathematicalObject)
    maps_to(function: Function, source: MathematicalObject, target: MathematicalObject)
    models(model: Model, target: MathematicalObject)
    approximates(model: Model, target: MathematicalObject)
    contradicts(a: Proposition, b: Proposition, context: string)

  constraints:
    - "Every mathematical symbol and operation must have a defined type and scope."
    - "Function inputs must belong to the declared domain and outputs to the codomain."
    - "Proof validity is distinct from empirical confidence and model fit."
    - "A theorem is valid only relative to its stated axioms, definitions and assumptions."
    - "Numerical approximations must not be presented as exact equalities."
    - "Dimensional quantities must use compatible units before arithmetic comparisons."
    - "Do not infer equivalence from similarity of notation or terminology."
    - "Cross-domain mappings must preserve the claimed structure or disclose losses."
    - "Confidence lies in [0,1] and describes evidence quality, not mathematical truth."

  inference:
    rule proof_dependency:
      if proves(p,t) and verified(p) and all_premises_satisfied(p)
      then established_under(t, assumptions_of(p))
      note: "Requires proof checking or explicitly identified trusted premises."
    rule dependency_transitivity:
      if depends_on(a,b) and depends_on(b,c)
      then indirectly_depends_on(a,c)
      note: "Do not infer logical implication solely from dependency."
    rule function_composition:
      if maps_to(f,a,b) and maps_to(g,b,c) and composable(f,g)
      then maps_to(compose(g,f),a,c)
      note: "Check types, totality and domain restrictions."
    rule equivalence_transport:
      if equivalent_under(a,b,h) and property_invariant_under(p,h)
      then transport_property_conditionally(p,a,b,h)
      note: "Only properties invariant under the specified equivalence may transfer."

  SEMANTIC_EQUIVALENCE:
    - "Normalise terms against versioned mathematical vocabularies."
    - "Distinguish equality, isomorphism, equivalence, approximation and analogy."
    - "Preserve axioms, domain restrictions, units and notation conventions."

  CONFIDENCE:
    range: [0,1]
    factors: [source_reliability, formal_verification, reproducibility, numerical_error, model_assumptions]
    policy: "Do not convert confidence scores into proof of a proposition."

  VALIDATION:
    - "Check type correctness, well-formed expressions and declared assumptions."
    - "Verify proof references, premise satisfaction and dependency cycles."
    - "Check numerical precision, error bounds and dimensional consistency."
    - "Flag contradictory assertions within the same formal context."
    - "Retain unresolved conjectures and alternative models without false certainty."

  FLOW_HASH:
    algorithm: "SHA-256"
    input: "canonicalised ontology version, entities, rules, assumptions and provenance"
    note: "Compute at execution; no hash value is claimed here."

  UPDATE_MODEL:
    strategy: "versioned, provenance-preserving incremental update"
    steps: [ingest, normalise, type_check, validate, compare, reconcile, recompute_inferences, publish]
    rollback: true

  AI_CONSUMPTION:
    representation: "typed mathematical knowledge graph with proof and assumption links"
    outputs: [domain_taxonomy, formal_statements, proof_dependencies, cross_domain_mappings, uncertainty_annotations, provenance_records]
    safeguards: [no_unverified_proofs, no_unqualified_equivalence, preserve_assumptions, preserve_units, distinguish_exact_from_approximate]
}

END FLOW ALGEBRA