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MathematicsLogicEstablished

Axiom

An axiom, postulate, or assumption, is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments.

AxiomMathematical proofTheoremMathematical logicPythagorean theoremTrianglePolygonTrigonometryAngle

At a glance

Type
information
Disciplines 2
MathematicsLogic
Mental models 0
—
Role in the graph
Connector

Key signals

Cross-disciplinary reach
Disciplines
2
  • Mathematics
  • Logic
Evidence & development

Dependencies

What this concept builds on and what it makes possible — derived from the atlas’s dependency, causal and structural relations, not from every related edge.

Enables · leads to

Mathematical proofdepends onAxiomEstablished

Open Mathematical proof →

mathematical proof depends on axiom.

Mechanism: A proof rests on axioms: it derives a new truth by chaining logical steps from statements accepted without proof.

Sources:
Theoremdepends onAxiomEstablished

Open Theorem →

theorem depends on axiom.

Mechanism: A theorem ultimately depends on axioms: it is a truth derived by pure logic from the starting assumptions a system accepts.

Sources:

System context

Axiomis part ofMathematical logicEstablished

Open Mathematical logic →

axiom is part of mathematical logic.

Mechanism: Axioms are the foundation of mathematical logic: the self-evident starting statements from which all other truths are proved.

Sources:

Structural role & consequence

Interpreted from the current atlas graph — what the connections mean, not just how many there are.

  • Currently dark in the atlas: 3 of 3 of its relations lack claim-level evidence.

    atlas representation · Describes the current Thinking OS representation, not the state of the world.

  • All 3 of its relationships stay within its own discipline — a field-specific concept in the current atlas.

    structural · Structural graph analysis — not a claim of importance, causation or history.

0%

cross-field
3 within-field, 0 cross-field

0 of 3 relations carry evidence · concept has a verified source

Strengths & constraints

Constraints

  • Evidence coverage currently thin in the atlas — few of its relationships carry claim-level evidence. atlas representation

Conditions

  • Read structurally — most of its relationships carry no external evidence yet, so claims here are graph-derived. structural

Dependency radial

What this concept builds on (left) and what it makes possible (right) — derived from dependency and causal relations.

Mathematical proofTheoremAxiom◀ builds onenables ▶

What builds on this

2 concepts build on this directly, 2 in total, across 2 disciplines.

LogicMathematics

Structural downstream reach along dependency edges — not a claim of historical necessity.

Seen through each discipline

How this concept sits in each of its fields — derived from its real connections in the graph, not asserted.

Mathematics

Through this lens it connects to Mathematical proof, Theorem and Mathematical logic.

Axiom through the Mathematics lens →

Logic

Through this lens it connects to Mathematical proof, Theorem and Mathematical logic.

Axiom through the Logic lens →

Key dates

  1. c. 300 BCEFormalizationEuclid's Elements set out the axiomatic method from postulates and common notions. — MacTutor History of Mathematics Archive

Concepts that look related but are not yet connected here — candidates for a connection to reason about, not established links.

The scientific picture
  • Connects 3 other ideas across 2 disciplines.
  • Most of its connections are of the “Dependency” kind.

Derived from the graph’s real structure — observations, not a score.

Sources