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Research draft

theorem

vr.tr.theorem · INF.KNW

Let an agent explain theorems by statement, proof, history and use, distinguish related terms, and point to formal verification resources.

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 theorems by statement, proof, history and use, distinguish related terms, and point to formal verification resources.

A mathematical statement proved true by logical deduction from axioms and previously established results, such as the Pythagorean theorem or the fundamental theorem of arithmetic; related terms include lemma, corollary and proposition.

What it is for: Building mathematical knowledge.

It can be state and explain a theorem; outline or check a proof; distinguish lemma, corollary and conjecture; find formalised proofs.

Distinguishing features

Proved statement

Depends on axioms

Builds on earlier results

Can be formally verified

What it looks like

Not physical; statements and proofs in texts and formal libraries.

How it is recognised

Named results with proofs

Lemmas and corollaries

Conjectures are unproved

Related models

is a kind of - category

proposition

is established by - method

mathematical proof

is contrasted with - unproved statement

conjecture

is related to - foundation

axiom

In practice

Families and kinds

major named theorems

lemmas

corollaries

existence and classification theorems

formally verified theorems

Standards and regulation

No specific regulation

Failure modes and hazards

Misstating hypotheses

Misattributing theorems

Treating conjectures as proved

Also called

Johnson bounddifferentiation rulelemmapumping lemmaisomorphism theoremJacobi–Anger expansionSerre dualityNewton's inequalitiesunexpected hanging paradoxMertens' theoremsAbelian theoremTauberian theoremBierlein extension theoremnetwork theoremfixed-point theoremGreen's identitiesWeyl's inequalityhypergeometric identitySiegel identityvaluative criterionproof of impossibilitypolarization identityAbelian and tauberian theoremsmetatheoremuniversal coefficient theoremFontené theoremstangent half-angle formulano-go theoremrelationship among probability distributionsspherical law of cosinesSahlqvist formulaMinkowski–Steiner formulaeconomic theoremfundamental theoremCassini and Catalan identitiesbase change theoremClarkson's inequalitiesclassification theoremMac Lane's coherence theoremGeneric flatness

+17

Where this came from

wikidata · CC0 1.0

Source repository Registry entry point

Also registered as vr.tr.theorem

Drafted structure

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

Statement What it says.

Hypotheses matter.

State

Precise statement.

State

Statement.

  1. What exactly does the theorem state, including all hypotheses? definition
  2. What happens if a hypothesis is dropped? boundary

Meaning

Intuition.

Meaning

Intuition.

  1. How can the theorem be explained intuitively? action
  2. Which example illustrates it? action
Proof Why it is true.

Proofs justify.

Outline

Proof sketch.

Outline

Proof outline.

  1. What are the main steps of a standard proof? definition
  2. Which earlier results does it use? provenance

Check

Verifying.

Check

Checking proofs.

  1. Is this proposed proof correct, and where might it fail? boundary
  2. Is there a formalised proof in a proof assistant? provenance
Terms Related terms.

Terminology differs.

Lemma

Lemmas and corollaries.

Lemma

Lemmas and corollaries.

  1. Is this result a lemma, corollary or main theorem? definition
  2. Why is it labelled that way? definition

Conjecture

Unproved statements.

Conjecture

Conjectures.

  1. Has this statement actually been proved, or is it still a conjecture? provenance
  2. Which source confirms its status? provenance
History Origins.

Attribution matters.

Who

Discoverers.

Who

Attribution.

  1. Who first proved the theorem, according to historians of mathematics? provenance
  2. Is the naming historically accurate? boundary

Uses

Applications.

Uses

Applications.

  1. Where is the theorem used in mathematics or applications? provenance
  2. Which fields depend on it? provenance

What the second pass must settle

  • Should each theorem be a separate entry?
  • How should formal libraries be linked?
  • How should naming disputes be recorded?
Draft as JSON