{ "model": { "rank": 4578, "code": "thing-q178377", "model_id": "vr.tr.mathematical-induction", "name": "mathematical induction", "purpose": "Enable an agent to recognise mathematical induction, record its proof obligations, assess whether they are discharged, and choose justified next proof steps.", "family": "Thing Registry", "category": "Information and virtual systems", "status": "research-draft", "kind": "thing", "plane": "INF", "domain": "INF.KNW", "industry": "", "version": "", "url": "/models/thing/q178377/", "tier": 2, "score": 74, "payload": { "layer": "wikidata", "aliases": [ "proof by weak induction", "proof by strong induction", "transfinite induction" ], "aliasCount": 3, "merged": 3, "knownIn": 74, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": null, "source": { "dataset": "wikidata", "item": "Q178377", "url": "https://www.wikidata.org/wiki/Q178377", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-09T20:32:41Z", "providers": [ "Codex" ], "breadth": "recalled by Codex without web access - no source was read", "missingProviders": [], "pass": 2, "cost": { "grok": { "seconds": 18.6, "error": "Reading additional input from stdin...\nOpenAI Codex v0.153.4\n--------\nworkdir: R:\\02_PROJECTS\\02_Meta_Models_Platforms\\Ver.cy\\current\\thing-registry-backlog\nmodel: gpt-6-astra\nprovider: openai\napproval: never\nsandbox: read-only\nreasoning effort: none\nreasoning summaries: none\nsession id: 01a087de-85f9-7e83-8ccf-d1367071beea\n--------\nuser\nDescribe what is already known about one registered thing. Answer as JSON only, no prose around it.\n\nThing: mathematical induction\nSense to describe: (none recorded)\nDomain code: INF.MED\nAlso known as: (none)\n\n\nContext for this batch of 1001 things:\n# Batch 00", "usd": 0, "recall": true }, "codex": { "seconds": 58.9, "error": "Reading additional input from stdin...\nOpenAI Codex v0.153.4\n--------\nworkdir: R:\\02_PROJECTS\\02_Meta_Models_Platforms\\Ver.cy\\current\\thing-registry-backlog\nmodel: gpt-6-astra\nprovider: openai\napproval: never\nsandbox: read-only\nreasoning effort: none\nreasoning summaries: none\nsession id: 01a087de-8609-7db3-8fcb-61ebe8737923\n--------\nuser\nYou are drafting a Vercy meta-model for one registered thing. Answer as JSON only, no prose around it.\n\nThing: mathematical induction\nRegistry id: vr.tr.mathematical-induction\nPlane / domain: INF / INF.MED\nRegistry definition: (none recorded)\nNames folded into" } } }, "metaModel": { "id": "THING-Q178377", "registryId": "vr.tr.mathematical-induction", "name": "mathematical induction", "version": "0.1.0-research.1", "entryKind": "thing", "family": "Thing Registry", "domain": [ "INF.MED" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q178377/", "model": { "registry_id": "vr.tr.mathematical-induction", "name": "mathematical induction", "purpose": "Enable an agent to recognise mathematical induction, record its proof obligations, assess whether they are discharged, and choose justified next proof steps.", "definition": "Mathematical induction is a deductive proof method that establishes a proposition for every natural number from a specified starting value by proving it at that value and proving that its truth at any such number implies its truth at the next.", "scope_statement": "This model owns mathematical induction as a deductive proof principle and its application to propositions indexed by natural numbers, including ordinary and strong induction, with explicit boundaries to structural and general well-founded induction.", "in_scope": [ "The indexed proposition, natural-number domain, starting index and quantified parameters", "Ordinary and strong induction and the justification for choosing between them", "Base cases, induction hypotheses, inductive steps and their dependency coverage", "Validity conditions, unresolved proof obligations and common induction errors", "The relationship between a completed induction argument and its quantified conclusion" ], "out_of_scope": [ "Empirical or statistical induction from observations to probable generalisations", "Recursive definitions and algorithms except for their role in an induction argument", "Structural induction over syntax trees or other inductively generated objects as an independently modelled method", "General well-founded and transfinite induction beyond natural-number indexing", "The mathematical subject matter of the theorem being proved", "Proof-assistant infrastructure and general proof-document management" ], "distinguishing_features": [ "The argument proves a universally quantified proposition over an explicitly specified natural-number range; checking many examples alone does not qualify.", "An ordinary induction step derives P(k+1) from an admissible assumption P(k), with k arbitrary in the stated range.", "A strong induction step may assume P(j) for every admissible j up to k when deriving P(k+1), rather than assuming the desired result without an index restriction.", "Established starting cases and the transition rule together cover every index claimed by the conclusion.", "The method yields a deductive conclusion conditional on its premises and mathematical setting, rather than a confidence estimate based on observed regularities." ], "characteristics": [ { "name": "Indexed proposition", "kind": "relation", "unit_or_values": "Predicate P(n), with any additional parameters and quantifiers explicit", "why_it_matters": "The base cases, induction hypothesis and conclusion must refer to the same proposition." }, { "name": "Index domain and starting index", "kind": "category", "unit_or_values": "A specified natural-number range n ≥ n₀, with the convention for whether zero belongs to the natural numbers recorded", "why_it_matters": "Determines the first required cases and the extent of the conclusion." }, { "name": "Induction formulation", "kind": "category", "unit_or_values": "Ordinary induction; strong induction; explicitly justified natural-number variant", "why_it_matters": "Determines which earlier instances may be assumed during the inductive step." }, { "name": "Parameter quantifier placement", "kind": "category", "unit_or_values": "Parameters fixed throughout the induction; universally quantified inside P(n); explicitly constrained dependent parameters", "why_it_matters": "Controls whether the induction hypothesis can be instantiated at changed parameter values." }, { "name": "Base-case coverage", "kind": "state", "unit_or_values": "Not specified; required cases identified; partially proved; all required cases proved", "why_it_matters": "A valid transition cannot compensate for missing starting cases." }, { "name": "Inductive dependency", "kind": "relation", "unit_or_values": "The earlier indices or indexed statements used to prove the target instance", "why_it_matters": "Exposes unavailable assumptions, dependency gaps and circular reasoning." }, { "name": "Proof-obligation status", "kind": "state", "unit_or_values": "Unassessed; incomplete; invalid as written; discharged relative to stated premises", "why_it_matters": "Distinguishes an induction plan from an argument that supports its conclusion." }, { "name": "Verification basis", "kind": "category", "unit_or_values": "Unchecked argument; human-reviewed argument; mechanically checked argument in an identified formal system", "why_it_matters": "Records how the assessment was established without conflating mathematical validity with a particular checking procedure." } ], "affordances": [ "Extract an indexed proposition and generate its base-case and inductive-step obligations.", "Choose ordinary or strong induction by inspecting the earlier instances needed by the proposed step.", "Check that quantifiers and side conditions permit each use of the induction hypothesis.", "Detect missing initial cases, uncovered indices and circular dependencies.", "Propose a strengthened proposition or generalised parameter when the current induction hypothesis is insufficient.", "Issue the quantified conclusion only after the required obligations and index coverage are established." ] }, "sources": [], "structure": { "bundles": [ { "id": "indexed-claim", "name": "Indexed claim", "description": "The exact proposition and range to which induction is applied.", "rationale": "An induction argument can appear complete while proving a different proposition or a smaller range than intended.", "layers": [ { "id": "predicate-and-quantifiers", "name": "Predicate and quantifiers", "description": "Makes the induction variable and other variables logically explicit.", "findings": [ { "id": "stable-induction-predicate", "name": "Stable induction predicate", "description": "Record P(n) with its quantifier structure so that each proof stage can be checked against the intended claim.", "questions": [ { "text": "What exactly is P(n), including every side condition and quantifier?", "kind": "definition", "id": "stable-induction-predicate-q01" }, { "text": "Which parameters remain fixed, and which must be universally quantified inside P(n) for the step to work?", "kind": "boundary", "id": "stable-induction-predicate-q02" } ] } ] }, { "id": "natural-number-range", "name": "Natural-number range", "description": "Identifies the initial index and every index included in the intended conclusion.", "findings": [ { "id": "declared-index-coverage", "name": "Declared index coverage", "description": "Record the admissible indices explicitly rather than relying on an unstated convention about natural numbers.", "questions": [ { "text": "Does the claim begin at zero, one or another specified natural number?", "kind": "definition", "id": "declared-index-coverage-q01" }, { "text": "Does the intended conclusion cover all subsequent natural numbers or only a restricted subset requiring an additional argument?", "kind": "boundary", "id": "declared-index-coverage-q02" } ] } ] } ] }, { "id": "induction-principle", "name": "Induction principle", "description": "The rule that licenses passage from starting cases and conditional steps to a universal conclusion.", "rationale": "Different formulations grant different induction hypotheses and must not be mixed implicitly.", "layers": [ { "id": "formulation-selection", "name": "Formulation selection", "description": "Matches the induction formulation to the proof's dependency requirements.", "findings": [ { "id": "permitted-hypothesis", "name": "Permitted hypothesis", "description": "Record whether the step may use only P(k), every P(j) from the starting index through k, or another explicitly justified collection.", "questions": [ { "text": "Is the argument using ordinary induction, strong induction or a stated variant?", "kind": "definition", "id": "permitted-hypothesis-q01" }, { "text": "Which earlier instances does the proposed step need, and does the selected formulation permit all of them?", "kind": "action", "id": "permitted-hypothesis-q02" } ] } ] }, { "id": "logical-setting", "name": "Logical setting", "description": "Records how the induction principle is supplied in the mathematical setting being used.", "findings": [ { "id": "principle-justification", "name": "Principle justification", "description": "Identify the adopted induction rule, axiom instance or theorem when the argument requires formal grounding.", "questions": [ { "text": "Which stated rule, axiom schema or established theorem licenses this induction argument?", "kind": "provenance", "id": "principle-justification-q01" }, { "text": "Does the chosen formal setting impose restrictions on the predicates to which that principle applies?", "kind": "boundary", "id": "principle-justification-q02" } ] } ] } ] }, { "id": "base-and-step-obligations", "name": "Base and step obligations", "description": "The mathematical work required to establish the starting cases and the general transition.", "rationale": "A named induction method supplies no conclusion until its concrete obligations have been proved.", "layers": [ { "id": "initial-cases", "name": "Initial cases", "description": "Determines and verifies the starting cases required by the transition.", "findings": [ { "id": "sufficient-starting-cases", "name": "Sufficient starting cases", "description": "Record each required initial instance and its proof, including extra instances needed by transitions using several predecessors.", "questions": [ { "text": "Which initial cases are required before the proposed transition can establish subsequent cases?", "kind": "definition", "id": "sufficient-starting-cases-q01" }, { "text": "Has each required case been established without invoking an unproved instance of the induction claim?", "kind": "action", "id": "sufficient-starting-cases-q02" } ] } ] }, { "id": "general-transition", "name": "General transition", "description": "Checks the conditional derivation at an arbitrary admissible index.", "findings": [ { "id": "valid-successor-derivation", "name": "Valid successor derivation", "description": "Record the induction hypothesis, target instance and derivation, including conditions needed for intermediate operations.", "questions": [ { "text": "Starting from the permitted induction hypothesis, what establishes P(k+1) for arbitrary admissible k?", "kind": "action", "id": "valid-successor-derivation-q01" }, { "text": "Are all required side conditions, such as nonzero divisors or valid predecessor indices, established throughout the step?", "kind": "boundary", "id": "valid-successor-derivation-q02" } ] } ] } ] }, { "id": "dependency-and-coverage", "name": "Dependency and coverage", "description": "Whether the proof reaches its entire claimed range using only justified dependencies.", "rationale": "Locally plausible steps can conceal circular assumptions or leave whole classes of indices unproved.", "layers": [ { "id": "hypothesis-use", "name": "Hypothesis use", "description": "Audits every appeal to an earlier indexed instance.", "findings": [ { "id": "admissible-instantiation", "name": "Admissible instantiation", "description": "Connect each use of the induction hypothesis to an allowed index and compatible parameter values.", "questions": [ { "text": "For each appeal to the induction hypothesis, which index and parameter values are substituted?", "kind": "definition", "id": "admissible-instantiation-q01" }, { "text": "Does any appeal require the target instance, a later instance or a parameter substitution not licensed by the hypothesis?", "kind": "boundary", "id": "admissible-instantiation-q02" } ] } ] }, { "id": "reachability-of-indices", "name": "Reachability of indices", "description": "Checks that established starting cases and the actual transition cover the claimed index domain.", "findings": [ { "id": "no-uncovered-index-classes", "name": "No uncovered index classes", "description": "Identify coverage gaps, such as proving only even indices with a step of two and a single even base case.", "questions": [ { "text": "Can every claimed index be reached from an established base case through the proved transitions?", "kind": "boundary", "id": "no-uncovered-index-classes-q01" }, { "text": "If the transition advances by more than one, which additional starting cases or reindexing argument are needed?", "kind": "action", "id": "no-uncovered-index-classes-q02" } ] } ] } ] }, { "id": "assessment-and-repair", "name": "Assessment and repair", "description": "The supported conclusion, remaining obligations and justified ways to revise an unsuccessful induction.", "rationale": "An agent must distinguish a false claim from an inadequate induction attempt and preserve that distinction when acting.", "layers": [ { "id": "completion-assessment", "name": "Completion assessment", "description": "Separates verified obligations from unsupported assertions and records the basis of review.", "findings": [ { "id": "conclusion-warrant", "name": "Conclusion warrant", "description": "Record the precise universal statement supported by discharged obligations and the premises on which it depends.", "questions": [ { "text": "Which obligations remain unresolved, and what exact conclusion is warranted by those already discharged?", "kind": "action", "id": "conclusion-warrant-q01" }, { "text": "What review or formal checking supports the recorded completion status?", "kind": "provenance", "id": "conclusion-warrant-q02" } ] } ] }, { "id": "failed-attempt-repair", "name": "Failed attempt repair", "description": "Classifies failures and identifies changes that could produce a valid argument.", "findings": [ { "id": "claim-or-method-revision", "name": "Claim or method revision", "description": "Distinguish a counterexample to the claim from missing base cases, an insufficient hypothesis or an invalid derivation.", "questions": [ { "text": "Does the failure exhibit a counterexample to P(n), or only a gap in this attempted proof?", "kind": "boundary", "id": "claim-or-method-revision-q01" }, { "text": "Would strengthening P(n), generalising a parameter, changing the starting index or using strong induction resolve the identified gap?", "kind": "action", "id": "claim-or-method-revision-q02" } ] } ] } ] }, { "id": "evidence-and-external-alignment", "name": "Evidence and external alignment", "description": "What the world already says about this thing, gathered so the model can be checked against it.", "rationale": "A model that cannot be lined up against existing standards, identifiers and practice cannot be adopted by anyone who already uses them.", "layers": [ { "id": "reported-evidence", "name": "Reported evidence", "description": "Findings from the breadth pass, kept separate from the structural claims.", "findings": [ { "id": "evidence-confidence-notes", "name": "Check these first", "description": "Recalled without web access and unsourced; every item is a lead to verify.", "evidence": [ "No sense was supplied; this describes the mathematical proof method, not a particular written exposition or creative work.", "Structural, well-founded and transfinite induction are generalizations of the natural-number method; the listed kinds overlap rather than forming disjoint categories.", "The starting natural number may be 0 or 1 by convention; formal treatments also distinguish induction as an axiom schema from a single second-order axiom." ], "questions": [ { "id": "evidence-confidence-notes-q01", "text": "Which of these check these first hold for the sense of mathematical induction this model covers, and on what evidence?", "kind": "provenance" } ] }, { "id": "evidence-kinds", "name": "Kinds and varieties", "description": "Recalled without web access and unsourced; every item is a lead to verify.", "evidence": [ "Ordinary (weak) induction", "Strong (complete) induction", "Simultaneous (mutual) induction", "Structural induction", "Well-founded induction", "Transfinite induction" ], "questions": [ { "id": "evidence-kinds-q01", "text": "Which of these kinds and varieties hold for the sense of mathematical induction this model covers, and on what evidence?", "kind": "provenance" } ] }, { "id": "evidence-real-world-use", "name": "Real-world use", "description": "Recalled without web access and unsourced; every item is a lead to verify.", "evidence": [ "Proving formulas for finite sums, inequalities and divisibility properties", "Establishing properties of recursively defined sequences", "Proving correctness of recursive algorithms", "Proving properties of syntax trees and other inductively defined structures", "Establishing invariants across successive computation steps" ], "questions": [ { "id": "evidence-real-world-use-q01", "text": "Which of these real-world use hold for the sense of mathematical induction this model covers, and on what evidence?", "kind": "provenance" } ] }, { "id": "evidence-failure-modes-and-hazards", "name": "Failure modes and hazards", "description": "Recalled without web access and unsourced; every item is a lead to verify.", "evidence": [ "Omitting a necessary base case or choosing a base case that does not cover the claimed domain", "Assuming the conclusion for the next case instead of deriving it from the induction hypothesis", "Proving an inductive step that fails at a boundary or skips cases", "Using an induction hypothesis too weak to support the required step", "Applying generalized induction without establishing that the underlying relation is well-founded" ], "questions": [ { "id": "evidence-failure-modes-and-hazards-q01", "text": "Which of these failure modes and hazards hold for the sense of mathematical induction this model covers, and on what evidence?", "kind": "provenance" } ] }, { "id": "evidence-neighbours", "name": "Neighbouring kinds and how to tell them apart", "description": "Recalled without web access and unsourced; every item is a lead to verify.", "evidence": [ { "name": "Inductive reasoning", "difference": "Inductive reasoning extrapolates from observations; mathematical induction is deductive and establishes every case covered by its premises." }, { "name": "Recursion", "difference": "Recursion defines objects or computations through simpler cases; induction proves propositions about cases." }, { "name": "Well-ordering principle", "difference": "The well-ordering principle asserts that every nonempty subset of the natural numbers has a least element; induction supplies a proof principle, equivalent under suitable foundational assumptions." }, { "name": "Proof by infinite descent", "difference": "Infinite descent rules out a counterexample by deriving a strictly smaller one indefinitely; ordinary induction proceeds from a base case through a successor implication." }, { "name": "Proof by exhaustion", "difference": "Exhaustion checks every case individually in a finite domain; induction uses a general step to cover an unbounded sequence of cases." } ], "questions": [ { "id": "evidence-neighbours-q01", "text": "Which of these neighbouring kinds and how to tell them apart hold for the sense of mathematical induction this model covers, and on what evidence?", "kind": "provenance" } ] } ] } ] } ] }, "openQuestions": [ "Does the existing Vercy catalogue already contain an authoritative model of mathematical induction or a proof-method model to which this registry entry should link?", "Should structural and well-founded induction be represented as neighbouring models or included as explicitly distinguished variants within this registry entry?", "Which authoritative sources should ground the published treatment of ordinary and strong induction and their relationship?", "How much formal-system detail should agents be required to record for informal proofs versus mechanically checked proofs?", "Should simultaneous induction and induction with non-unit steps receive dedicated layers, or be represented through explicitly justified variants and dependency coverage?" ], "statistics": { "bundles": 6, "layers": 11, "findings": 15, "questions": 25 } }, "draft": { "generator": "vr.draft.v3", "status": "draft-generated", "researched": false, "archetype": "discipline or field of knowledge", "method": "Written from the archetype playbook - what this kind of thing needs beyond identity and provenance - and from the structure that recurred across 6,333 models already researched by two engines. Applied to this entry by rule. No source was read for this thing and no claim here is researched. This entry carries no facets of its own, so they were inferred from its domain - a guess about a whole domain applied to one thing.", "facetsInferred": true, "nextPass": "A researcher replaces this draft with a sourced specification. Treat every sentence below as a proposal to argue with.", "purpose": "Give an agent a durable, checkable way to recognise a mathematical induction, record what state it is in, and decide what may be done with it.", "whatItIs": "Enable an agent to recognise mathematical induction, record its proof obligations, assess whether they are discharged, and choose justified next proof steps.", "characteristics": { "substance": "information", "origin": "conceptual", "agency": "inert" }, "whatYouCanDoWithIt": [ "read and interpreted" ], "distinguishingFeatures": [ "Names folded into this entry, which a task may need to split apart again: proof by weak induction, proof by strong induction, transfinite induction.", "3 finer distinctions are held as aliases rather than separate entries, because telling them apart needs a task that asks for it.", "Described in 74 Wikipedia languages, which is a measure of how widely the thing is known, not of how important it is." ], "openQuestionsForResearch": [ "Which of the bundles below does a real task actually need, and which are ceremony?", "What does this thing have that the facets do not capture at all?", "Which neighbouring kind is most often confused with a mathematical induction, and on what evidence are they told apart?" ], "whatItIsMadeOf": "content that has to be carried by something else", "physicalCharacter": [ "Does nothing on its own; everything it does, something else did to it.", "These come from the domain this entry sits in rather than from the entry itself, so treat them as a first guess about the whole domain applied to one thing." ], "whatCanBeDoneWithIt": [ "read it and act on what it says" ], "howItIsRecognised": [], "relatedModels": [ { "relation": "covers", "note": "Finer kinds folded into this entry because telling them apart needs a task that asks for it. Each is a model waiting to be split out when one does.", "targets": [ "proof by weak induction", "proof by strong induction", "transfinite induction" ] } ], "standing": "Described in 74 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is. 3 finer distinctions are held inside this entry as names rather than as separate models.", "structure": { "bundles": [ { "id": "identity-and-classification", "name": "Identity, naming and classification", "description": "How an agent tells one mathematical induction from another, and a mathematical induction from things that resemble it.", "rationale": "Recognition comes before every other claim. Without stable identity nothing else in the model can be trusted to be about the same thing twice.", "layers": [ { "id": "naming-and-identifiers", "name": "Names and identifiers", "description": "The names this thing goes by and the identifiers that survive translation and time.", "findings": [ { "id": "preferred-name-and-aliases", "name": "Preferred name, aliases and local names", "description": "Which name to use, which names mean the same thing, and which merely sound similar.", "questions": [ { "id": "preferred-name-and-aliases-q01", "text": "What identifies and describes the name of a mathematical induction, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "preferred-name-and-aliases-q02", "text": "Who or what asserted this about the name of a mathematical induction, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "preferred-name-and-aliases-q03", "text": "What may an agent decide or do once the name of a mathematical induction is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] }, { "id": "stable-identifiers", "name": "Stable identifiers and external keys", "description": "Identifiers that keep pointing at this kind of thing across systems and languages.", "questions": [ { "id": "stable-identifiers-q01", "text": "What identifies and describes an identifier for a mathematical induction, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "stable-identifiers-q02", "text": "Who or what asserted this about an identifier for a mathematical induction, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "stable-identifiers-q03", "text": "What may an agent decide or do once an identifier for a mathematical induction is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "classification-and-granularity", "name": "Classification and granularity", "description": "Where a mathematical induction sits among kinds, and how finely a task needs to cut it.", "findings": [ { "id": "kind-and-parents", "name": "Kind, parents and neighbouring kinds", "description": "The classes this thing belongs to and the ones it is next to.", "questions": [ { "id": "kind-and-parents-q01", "text": "What identifies and describes the kind of a mathematical induction, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "kind-and-parents-q02", "text": "Who or what asserted this about the kind of a mathematical induction, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "kind-and-parents-q03", "text": "What may an agent decide or do once the kind of a mathematical induction is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] }, { "id": "distinguishing-features", "name": "Distinguishing features", "description": "What separates a mathematical induction from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a mathematical induction, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "distinguishing-features-q02", "text": "Who or what asserted this about what distinguishes a mathematical induction, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "distinguishing-features-q03", "text": "What may an agent decide or do once what distinguishes a mathematical induction is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "state-and-lifecycle", "name": "State and lifecycle", "description": "The states a mathematical induction passes through and the events that move it between them.", "rationale": "Most decisions about a thing depend on what state it is in now, which is a claim with a time on it, not a property.", "layers": [ { "id": "lifecycle-stages", "name": "Lifecycle stages", "description": "From coming into existence to ceasing to be one of these.", "findings": [ { "id": "stages-and-transitions", "name": "Stages and transitions", "description": "The stages worth naming and what moves a mathematical induction between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a mathematical induction, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "stages-and-transitions-q02", "text": "Who or what asserted this about the lifecycle of a mathematical induction, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "stages-and-transitions-q03", "text": "What may an agent decide or do once the lifecycle of a mathematical induction is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "observations-and-status", "name": "Observations and current status", "description": "What is observed about a mathematical induction, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a mathematical induction is recorded so that it can be superseded rather than overwritten.", "questions": [ { "id": "observation-record-q01", "text": "What identifies and describes an observation of a mathematical induction, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "observation-record-q02", "text": "Who or what asserted this about an observation of a mathematical induction, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "observation-record-q03", "text": "What may an agent decide or do once an observation of a mathematical induction is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "content-and-access", "name": "Content, encoding and access", "description": "What a mathematical induction says, how it is encoded and who may read it.", "rationale": "An informational thing carries content that can be copied, versioned and withheld, none of which its physical carrier explains.", "layers": [ { "id": "content-and-encoding", "name": "Content and encoding", "description": "The content itself, its format and its language.", "findings": [ { "id": "content-and-format", "name": "Content, format and language", "description": "What a mathematical induction contains and in what form it is held.", "questions": [ { "id": "content-and-format-q01", "text": "What identifies and describes the content of a mathematical induction, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "content-and-format-q02", "text": "Who or what asserted this about the content of a mathematical induction, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "content-and-format-q03", "text": "What may an agent decide or do once the content of a mathematical induction is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "access-and-rights", "name": "Access, rights and retention", "description": "Who may read, copy or change it, and for how long it is kept.", "findings": [ { "id": "access-rules", "name": "Access rules and retention", "description": "The permissions attached to a mathematical induction and the period it survives.", "questions": [ { "id": "access-rules-q01", "text": "What identifies and describes access to a mathematical induction, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "access-rules-q02", "text": "Who or what asserted this about access to a mathematical induction, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "access-rules-q03", "text": "What may an agent decide or do once access to a mathematical induction is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "subject-and-method", "name": "Subject matter and method", "description": "What mathematical induction studies and how it establishes anything.", "rationale": "A field is identified by its questions and its methods, and those are what distinguish it from the fields next to it.", "layers": [ { "id": "subject", "name": "Subject matter", "description": "The questions it takes as its own.", "findings": [ { "id": "subject-record", "name": "Questions and objects of study", "description": "What it is about, stated so a neighbouring field can be told apart.", "questions": [ { "id": "subject-record-q01", "text": "What questions and objects does mathematical induction take as its own?", "kind": "definition" }, { "id": "subject-record-q02", "text": "Which questions does it share with a neighbouring field, and who claims them?", "kind": "boundary" } ] } ] }, { "id": "method", "name": "Method and evidence", "description": "How claims are established and what counts as evidence.", "findings": [ { "id": "method-record", "name": "Methods and standards of evidence", "description": "The methods used and what they are taken to establish.", "questions": [ { "id": "method-record-q01", "text": "By what methods does mathematical induction establish claims, and what counts as sufficient evidence?", "kind": "provenance" }, { "id": "method-record-q02", "text": "What kind of claim can this field not settle, and where should an agent look instead?", "kind": "action" } ] } ] } ] }, { "id": "institutions-and-classification", "name": "Institutions and classification", "description": "Where mathematical induction is practised and how catalogues place it.", "rationale": "The institutional footprint is the evidence that a field exists as a field rather than as a topic.", "layers": [ { "id": "institutions", "name": "Institutional markers", "description": "Societies, journals, degrees and departments.", "findings": [ { "id": "institution-record", "name": "Bodies and venues", "description": "Where the field organises itself.", "questions": [ { "id": "institution-record-q01", "text": "Which societies, journals or degrees mark mathematical induction as an established field?", "kind": "provenance" }, { "id": "institution-record-q02", "text": "What would show that it is emerging or dissolving rather than established?", "kind": "boundary" } ] } ] }, { "id": "classification", "name": "Classification schemes", "description": "Where library and research classifications put it.", "findings": [ { "id": "classification-record", "name": "Codes and their disagreements", "description": "The classification codes that place it, and where they differ.", "questions": [ { "id": "classification-record-q01", "text": "Which classification schemes place mathematical induction, under what codes?", "kind": "definition" }, { "id": "classification-record-q02", "text": "Where do those schemes disagree about its scope?", "kind": "boundary" } ] } ] } ] }, { "id": "provenance-and-evidence", "name": "Provenance, evidence and time", "description": "Where every claim about a mathematical induction came from and when it held.", "rationale": "A claim without a source and a time cannot be superseded, only overwritten, and an agent that overwrites loses the ability to explain itself.", "layers": [ { "id": "source-and-authority", "name": "Source and authority", "description": "Who said it, on what evidence, and how strongly.", "findings": [ { "id": "claim-provenance", "name": "Claim provenance and confidence", "description": "The authority behind each claim about a mathematical induction and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a mathematical induction, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "claim-provenance-q02", "text": "Who or what asserted this about a claim about a mathematical induction, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "claim-provenance-q03", "text": "What may an agent decide or do once a claim about a mathematical induction is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "time-and-versions", "name": "Time, versions and supersession", "description": "When a claim was true, when it was learnt, and what replaced it.", "findings": [ { "id": "validity-and-supersession", "name": "Validity period and supersession", "description": "How an old claim about a mathematical induction is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a mathematical induction, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "validity-and-supersession-q02", "text": "Who or what asserted this about the validity of a claim about a mathematical induction, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "validity-and-supersession-q03", "text": "What may an agent decide or do once the validity of a claim about a mathematical induction is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] } ] }, "statistics": { "bundles": 6, "layers": 12, "findings": 14, "questions": 38 } } } }