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A counterexample in logic, philosophy, or mathematics is a demonstration, display, or finding which establishes either: (1) an exception contrary to a proposed general rule, hypothesis, or axiom, or (2) a refutation of a proposed argument.


  • Counterexample philosophy is a distinctive pattern of argumentation philosophers since Plato have employed when attempting to hone their conceptual tools.
  • Occasionally, one individual may come up with a "proof," and another with a "counterexample." Since a valid proof and counterexample cannot peacefully coexist, either the proof has some logical or mathematical flaw, or the counterexample does not faithfully represent the conditions involved, or perhaps both. This is another reason why it is so important to have good command of the underlying logic.
  • Whenever the bigger theorems are stated and proven, Landau usually shows that all the hypotheses are needed by dropping each one and giving a counterexample. In some cases the counterexamples are very elaborate, such as van der Waerden’s continuous, nowhere differentiable function, and a continuous function whose Fourier series diverges.

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