math-notes/logic-proofs/01-what-is-a-proof.md

What is a Proof?

Source: MIT 6.1200J Lecture 01

Definition

A mathematical proof is a verification of a proposition by a chain of logical deductions from a base set of axioms.

Key Components

1. Proposition — A statement that is either True or False
2. Axioms — Assumed-to-be-true base statements
3. Logical deductions — Chain of reasoning connecting axioms to conclusion

Important: Different Proof Contexts

• Physics: Experiment/observation
• Statistics: Sampling
• Law: Judge/jury verdict
• Business: Authority
• Mathematics: Logical deduction from axioms

Why State Axioms?

• Mathematics requires assumptions (axiom = stated assumption)
• Different axiom systems lead to different mathematical worlds
• Example: Euclidean vs. Hyperbolic geometry — both valid with different parallel axioms

Related

• ../discrete-math/00-index
• 02-propositions