Definition
Plain language
A single case that shows a general claim is false.
As stated in the literature
An instance that violates a universally-quantified statement, disproving it; used in formal review to demonstrate a missing hypothesis in a theorem as stated.
Also called: counterexamples
Why it matters: One well-chosen counterexample can collapse a sweeping claim, which is why it's a sharp tool for catching flawed reasoning or missing assumptions.
For example, the claim 'all swans are white' is undone the moment someone points to a single black swan.
Heard on the show
“" Mathematical knowledge advances through a cycle of conjecture, counterexample, and revision, not through clean accumulation of theorems.”Episode 029 — Why Forty-Eight Percent on FrontierMath Isn't the Real Story in DeepMind's New Math Paper