Week 10: Proof by elimination

Occam's razor, reductio ad absurdum, and a mathematics tradition that treats the shortest correct proof as a mark of taste, not just economy

26 April 2027

A reductio doesn’t prove its conclusion by building toward it; it proves by ruling out the alternative until only one option can still be standing. Mathematicians talk about an “elegant” proof the way a film editor talks about a clean cut — fewer moves, nothing carried that isn’t load-bearing — and Occam’s razor is the same instinct promoted to a general principle: prefer the account that needs the fewest assumptions, not because simpler is prettier, but because every extra assumption is one more thing that could be wrong.

Paul Erdős’s notion of “The Book” — an imagined volume God keeps of the most elegant proof of every theorem — makes explicit what this course has been circling since week 1: a discipline can have a shared, largely tacit standard for which cut was the right one, argued about, sometimes overturned, and taken completely seriously by people who’d never call it taste.

Outline#

• reductio ad absurdum: proving by elimination rather than construction
• Occam’s razor as elimination promoted to a general principle
• Erdős’s Book, and mathematics’s own version of “taste is what you leave out”