the.com/counterexample
one stubborn case that drags a sweeping claim out behind the barn.
means A specific case that disproves a general claim or proposed rule by satisfying its conditions yet failing its conclusion.
from A transparent compound: "counter-" (against, from Latin "contra") welded onto "example" (from Latin "exemplum," a sample taken out, from "eximere," to take out). Literally a sample held up against an assertion. It's a workhorse of mathematics and logic, where a single counterexample is enough to topple a universal claim — the technical use crystallizing in 19th- and 20th-century formal reasoning.
- kills proof
- A single instance topples any universal statement
- math weapon
- Disproving needs one; proving needs all
- famous slayer
- Frege's logic crumbled to Russell's one paradox
- asymmetry
- Hard to confirm rules, easy to break them
russell's paradox — set theory counterexample: the set of all sets that don't contain themselves, proved frege's axioms inconsistent in 1901
non-euclidean geometry — counterexample to euclid's parallel postulate: spherical geometry on earth's surface where parallel lines meet
weierstrass function — continuous but nowhere differentiable function, counterexample to belief that continuity implies differentiability (1872)