Counterexample

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.

Quotes

• Counterexample philosophy is a distinctive pattern of argumentation philosophers since Plato have employed when attempting to hone their conceptual tools.
  • Michael A. Bishop: (1992). "The possibility of conceptual clarity in philosophy". American Philosophical Quarterly 29 (3): 267-277.

• 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.
  • John F. Lucas: Introduction to Abstract Mathematics (2nd ed.). Rowman & Littlefield. 1990. p. 97. ISBN 978-0-912675-73-2.  (1st edition, 1985) (The author is not the famous philosopher John Randolph Lucas.)

• 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.
  • Allen Stenger: Review of Differential and Integral Calculus by Edmund Landau. MAA Reviews, Mathematical Association of America (June 12, 2009). (The book reviewed is the 2001 translation of Landau's Einführung in die Differential- und Integralrechnung published by Noordhoff in 1934.)

• What is the role of counterexamples in mathematics? (Are there any in Euclid?)
  • Albert Wilansky: (1983). "Book Review: Counterexamples in topological vector spaces by S. M. Khaleelulla". Bulletin of the American Mathematical Society 8 (2): 389–391. ISSN 0273-0979. DOI:10.1090/S0273-0979-1983-15138-8.