在数学里，有着许多明显矛盾的虚假证明存在。即使其证明是有缺陷的，其错误－通常是经过设计的－却常是较难抓摸的。这些谬误一般都尽止于好奇而已，但可以被使用显示严谨在数学中的重要性。

大多数此类的证明都仰赖着同种错误的变形。此一错误为采一非单射的函数f，以观察对某些x和y，会有f(x)=f(y)，来（错误地）做出x=y的结论。零除数是此类错误的一特例；f为将x映射至x×0的函数，而其错误的一步是起于将x×0=y×0的等式做成x=y的结论。相似地，下面证明了5=4的句子也是以函数f(x)=x的同一种错误造成的。其错误的一步始于有某个x和y会使得x=y的一正确申论，然后做出了x=y的一错误结论。

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.