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Research draft

mathematical proof

vr.tr.mathematical-proof · INF.KNW

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

Research draft, second pass

A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.

written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify

Researched by: Claude

Purpose and description

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.

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.

What it is for: Rigorous arguments establishing mathematical truths.

It can be explain proof and techniques; construct and check proofs; describe formal proof; support learning.

Distinguishing features

Deductive rigour

Techniques

Formalisable

Peer review

What it looks like

Not physical; written arguments.

How it is recognised

Deductive argument from axioms

Establishes a theorem

Evidence and heuristics are not proofs; a conjecture is unproved

Related models

is a kind of - category

argument

is a kind of - category

mathematical reasoning

is related to - statements often proved

inequality

is related to - objects proofs concern

imaginary

In practice

Families and kinds

direct proofs

proofs by contradiction and contrapositive

proofs by induction

constructive and existence proofs

computer-assisted and formal proofs

Standards and regulation

Conventions of mathematical writing

Peer review in journals

Proof assistant standards

Failure modes and hazards

Circular reasoning and gaps

Confusing examples with proofs

Errors in long proofs

Also called

computer-assisted proofstrategy-stealing argumentproof by contrapositiveProofs of Fermat's theorem on sums of two squaresindirect proofproof of the pythagorean theoremexponent combination lawswell-founded inductionnon-constructive proofproof by contradictionnon-surveyable prooftermination proofproofs of minimax theoremdirect proofcombinatorial proofproofs of Fermat's little theoremanalytic proofinductive stepCantor's diagonal argumentstatistical proofumbral moonshineprobabilistically checkable proofproof that e is irrationalproof by exhaustionThe Fermat method

Where this came from

wikidata · CC0 1.0

Source repository Registry entry point

Drafted structure

Bundle to layer to finding to question, as the second pass will find it: 4 bundles · 8 layers · 8 findings · 16 questions.

Understand What proofs are.

Mathematics.

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.

Practice.

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.

Technology.

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.

Education.

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

What the second pass must settle

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