Mathematical fallacy

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.

Soundtrack

A soundtrack, also written sound track, can be recorded music accompanying and synchronized to the images of a motion picture, book, television program or video game; a commercially released soundtrack album of music as featured in the soundtrack of a film or TV show; or the physical area of a film that contains the synchronized recorded sound.

Origin of the term

In movie industry terminology usage, a sound track is an audio recording created or used in film production or post-production. Initially the dialogue, sound effects, and music in a film each has its own separate track (dialogue track, sound effects track, and music track), and these are mixed together to make what is called the composite track, which is heard in the film. A dubbing track is often later created when films are dubbed into another language. This is also known as a M & E track (music and effects) containing all sound elements minus dialogue which is then supplied by the foreign distributor in the native language of its territory.