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mathematical proof

vr.tr.mathematical-proof · thing-q11538

Let an agent explain what a mathematical proof is and the main proof techniques, help construct, check and explain proofs, describe formal and computer-assisted proof, and support learning proof writing.

Thing Registry Information and virtual systems INF.KNW
Bundle → Layer → Finding → QuestionsClassifiersWhat it isWhy it existsDistinguishing featuresWhat robots and AI may and may not doMoral aspectsOwnersLinks to other meta-modelsIdentity and identifiersDirect propertiesRecognitionCapabilities and actionsHazards and failure modesStandards and interfacesContext of useSources

Bundle → Layer → Finding → Questions Filled

4 bundles · 8 layers · 8 findings · 16 questions

Understand What proofs are.

Concept

Concept and standards.

Concept

Concept.

  1. What makes an argument a mathematical proof, and how do axioms, definitions and inference rules function in it? definition
  2. Which entry fits logic? action

Techniques

Techniques.

Techniques

Techniques.

  1. How do direct proof, contradiction, contrapositive, induction and construction work, and when is each used? definition
  2. Which entry fits a specific technique? action
Write Constructing and checking.

Construct

Constructing a proof.

Construct

Constructing.

  1. How can this statement be proved, and which technique fits? action
  2. Is the statement precisely stated? boundary

Check

Checking a proof.

Check

Checking.

  1. Is this proof valid, and where are its gaps or errors? action
  2. Which entry fits common proof errors? action
Formal Formal and computer proof.

Assistants

Proof assistants.

Assistants

Assistants.

  1. How do proof assistants and formal verification work, and which proofs have been formalised? provenance
  2. Which entry fits a specific proof assistant? action

Computer

Computer-assisted proofs.

Computer

Computer.

  1. What are computer-assisted proofs, and what debates surround their acceptance, with positions attributed? provenance
  2. Which references are standard? provenance
Learn History and teaching.

History

History.

History

History.

  1. How did the concept of proof develop from Euclid to modern foundations, and which proofs are famous? provenance
  2. Which entry fits a famous proof? action

Teach

Teaching proof.

Teach

Teaching.

  1. How can proof writing be taught to beginners? action
  2. Which misconceptions arise? provenance

Classifiers Filled

Family
Thing Registry
Category
Information and virtual systems
Entry kind
thing
Plane
INF
Domain
INF.KNW
Other names and narrower kinds
computer-assisted proof, strategy-stealing argument, proof by contrapositive, Proofs of Fermat's theorem on sums of two squares, indirect proof, proof of the pythagorean theorem, exponent combination laws, well-founded induction, non-constructive proof, proof by contradiction, non-surveyable proof, termination proof

What it is Filled

A rigorous argument that establishes the truth of a mathematical statement from axioms, definitions and previously proved results by valid inference, in forms such as direct proof, proof by contradiction, proof by contrapositive, induction, construction, exhaustion, strategy-stealing arguments and computer-assisted proofs; proofs are the standard of certainty in mathematics, are written for human readers or checked by proof assistants, and famous examples include proofs of the Pythagorean theorem and of Fermat theorem on sums of two squares.

Why it exists Filled

Let an agent explain what a mathematical proof is and the main proof techniques, help construct, check and explain proofs, describe formal and computer-assisted proof, and support learning proof writing.

Distinguishing features Filled

  • Deductive rigour
  • Techniques
  • Formalisable
  • Peer review

What robots and AI may and may not do Filled

Must not

  • Present a set of examples or a numerical check as a proof.
  • Claim a proof is correct when it has not been checked step by step.
  • Attribute a proof or theorem to a mathematician without evidence.
  • Submit proofs on a student's behalf in assessments where that is not allowed.

Only with a human decision

  • Declaring a disputed proof of an open problem correct in a publication or award.

May

  • Check the steps of a proof and point out gaps, circular reasoning or unproved assumptions.
  • Explain common proof techniques such as induction or contradiction with examples.
  • Help formalise a proof in a proof assistant when the user asks.

Moral aspects Filled

  • Credit for proofs shapes careers; misattribution wrongs mathematicians.
  • Errors in proofs that others rely on, such as in safety or cryptography, can spread widely.

Who is affected

  • Mathematicians and students
  • Users of results built on the proof
  • Teachers and examiners

Owners Filled

Steward

Nobody owns a proof's truth; authors answer for their published proofs and journals for review.

Links to other meta-models Filled

parent

  • Q122766417 - registry parent class
  • Q186619 - registry parent class

related

  • argument - category
  • mathematical reasoning - category
  • inequality - statements often proved
  • imaginary - objects proofs concern

What else AI and robots need to interact with it Filled

Identity and identifiers required Filled

  • Vercy registry: vr.tr.mathematical-proof
  • Wikidata: Q11538 (https://www.wikidata.org/wiki/Q11538)

Direct properties not applicable Not applicable

Not applicable

Plane INF: no invented physical properties.

Recognition optional Filled

  • Deductive argument from axioms
  • Establishes a theorem
  • Evidence and heuristics are not proofs; a conjecture is unproved
  • Not physical; written arguments.

Capabilities and actions required Filled

  • explain proof and techniques
  • construct and check proofs
  • describe formal proof
  • support learning

Hazards and failure modes required Filled

  • Circular reasoning and gaps
  • Confusing examples with proofs
  • Errors in long proofs

Standards and interfaces required Filled

  • Conventions of mathematical writing
  • Peer review in journals
  • Proof assistant standards

Context of use required Filled

  • Rigorous arguments establishing mathematical truths.
  • direct proofs
  • proofs by contradiction and contrapositive
  • proofs by induction
  • constructive and existence proofs
  • computer-assisted and formal proofs

Sources Filled

  1. Wikidata item Q11538: mathematical proof - identity and sense of the item
  2. Wikipedia: Mathematical proof - general description of the item

Open questions

  • Should each technique be a separate entry?
  • How should proof assistants be linked?
  • How should famous proofs be linked?

Machine files

Card as JSON Specification Card backlog vercy.card/1

Provenance

thing registry research (pass 2) · unreviewed

Built from: models/things/publications/thing-q11538/spec.json