# 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|Discrete Math Foundations]]
- [[02-propositions|Propositions]]