In
mathematics, certain kinds of mistaken proof are often exhibited, and sometimes collected, as illustrations of a concept of
mathematical fallacy. There is a distinction between a simple
mistake and a
mathematical fallacy in a proof: a mistake in a proof leads to an
invalid proof just in the same way, but in the best-known examples of mathematical fallacies, there is some concealment in the presentation of the proof. For example, the reason validity fails may be a
division by zero that is hidden by algebraic notation. There is a striking quality of the mathematical fallacy: as typically presented, it leads not only to an absurd result, but does so in a crafty or clever way. Therefore, these fallacies, for pedagogic reasons, usually take the form of spurious
proofs of obvious
contradictions. Although the proofs are flawed, the errors, usually by design, are comparatively subtle, or designed to show that certain steps are conditional, and should not be applied in the cases that are the exceptions to the rules.