Proofs by Contradiction

PROOF BY CONTRADICTION (also called REDUCTIO AD ABSURDUM) is a powerful proof technique. Instead of proving X directly, you ASSUME NOT-X (the opposite). Then you derive a CONTRADICTION — something that cannot be true. Since assuming NOT-X leads to absurdity, NOT-X must be false. Therefore X is true. The technique works because in classical logic, statements are either true or false — no middle ground.

Famous examples. EUCLID proved that there are INFINITELY MANY PRIMES by assuming there are FINITELY many, then constructing a number that contradicts that assumption. The proof that √2 is IRRATIONAL: assume √2 = a/b in lowest terms. Square both sides: 2 = a²/b². So a² = 2b². So a is even. So a = 2k for some integer k. So 4k² = 2b², or b² = 2k². So b is also even. But a/b was supposed to be in lowest terms — contradiction. Therefore √2 cannot be written as a fraction.

When to use it. (1) When direct proof seems hard. (2) When the negation provides handles to grip. (3) For existence claims (showing something must exist). Some philosophers debate whether proof by contradiction always works — INTUITIONIST mathematicians question it. But classical logic accepts it as standard. Many of mathematics' most famous proofs use this technique.

Proof by contradiction is logic's aikido — using the opponent's strength against itself. A compact, powerful method that mathematicians and philosophers have used for over 2,000 years.