Where should I start learning mathematical proof?

This series is organized into four levels: Introduction, Basic, Intermediate, and Advanced. If you are not yet comfortable with symbolic logic, start with Introduction, where you will learn propositions, logical operations, conditional statements, and contrapositives. Then move on to Basic, which covers direct proof, proof by contradiction, mathematical induction, and other foundational techniques.

What is the difference between a "proof" and an "explanation"?

An "explanation" aims to convince through intuition or examples, while a "proof" is a procedure that derives a conclusion from assumptions using logical rules alone. A single counterexample invalidates a proof, so proofs demand both reproducibility and logical rigor.

What are the standard proof techniques to learn?

The main techniques are: direct proof (derive the conclusion from the assumptions), proof by contrapositive (show ¬Q ⇒ ¬P), proof by contradiction (negate the conclusion and derive a contradiction), mathematical induction (for statements about natural numbers), case analysis, and constructive vs. non-constructive existence proofs. The appropriate technique depends on the structure of the problem.

Proof

Mathematical Proofs

About this series

A mathematical proof is the activity of determining the truth of a proposition through logical reasoning. This series studies proof from its foundations through to advanced techniques in a stepwise manner.

Proof is a fundamental skill needed across every area of mathematics, and it also plays an important role in cultivating logical thinking.

Study by level

Introduction

Learn the foundations of propositions and logic.

What is a proposition?
Logical operations (AND, OR, NOT)
The meaning of "if … then"
Converse, inverse, contrapositive
Reading proofs

Basic

Master the standard proof techniques.

Direct proof
Proof by contradiction
Proof by contrapositive
Mathematical induction
Existence and uniqueness

Intermediate

Learn more advanced proof techniques.

Proofs involving quantifiers
Epsilon-delta arguments
Proofs of equivalence
Techniques of case analysis
Sets and logic

Advanced

Explore the foundations of mathematics and the limits of proof.

Constructive and non-constructive proofs
Well-ordering principle and infinite descent
Diagonal argument
Axiom of choice
The limits of proof (incompleteness theorems)

Reading · standalone articles

How Many Axioms Is High-School Mathematics Built From?

Almost all of high-school mathematics develops from the 15 axioms of the real numbers — number construction, completeness, coordinatized geometry, the place of logic, and the high-school/university divide.

Learning flow

Main topics

Propositional logic

Truth values, logical operations, implication, contrapositive — the language in which proofs are written.

Proof techniques

Direct proof, proof by contradiction, proof by contrapositive, mathematical induction, and other approaches.

Handling infinity

Epsilon-delta arguments, the well-ordering principle, the diagonal argument — techniques for reasoning about the infinite.

Foundations of mathematics

The axiom of choice, the incompleteness theorems, and other deep topics in the foundations of mathematics.

Study tips

Work it out by hand. Don't just read proofs — write them out yourself.
Ask "why". Always be aware of why each step is justified.
Look for counterexamples. Trying to find a case where the claim fails deepens understanding.
Generalize and specialize. Move back and forth between specific examples and general principles.