# Definition:Mistake

## Definition

A mistake (as opposed to an error) is an inaccuracy or (deduced) falsehood caused by (ultimately) human fault.

Synonyms include such terms as boob, bloomer, blooper, blunder, cock-up, bug (computer science), mess-up, oversight, howler and other less polite words frequently encountered at the backs of schoolrooms worldwide.

### Typo

A typo (abbreviation for typographical error) is a mistake arising from pressing the wrong button on the machine via which you are transcribing your thoughts onto a storage medium. These can be corrected by assiduous checking by the writer.

### Spello

A spello (abbreviation for spelling mistake) is a mistake caused by inadequate spelling skills of the writer. These can sometimes be corrected by the writer having the self-doubt necessary to occasion the act of referring to a reliable manual, for example, a dictionary. A competent proof-reader (i.e. one whose spelling skills are more comprehensive than the writer) is usually an adequate resource for reducing spelloes.

### Homophone Horror

A homophone horror is a mistake caused by the use of a word which sounds the same as, but is spelt differently to, the word intended. A proficient proof-reader is needed to reduce instances of this sort of mistake.

Both spelloes and typoes can be reduced by mechanical means, e.g. spell-checkers, but these are unreliable. Homophone horrors are not subject to being detected by a spell-checker, although a grammar-checker may catch one (if you're lucky). Devices which interpret semantic meaning may help here, but they are not widely available.

## Mapping Definition Mistakes

Here are some supposed definitions of mappings which contain mistakes.

### Image Elements not in Codomain

$f: \N \to \N$ defined as: $\forall x \in \N: x \mapsto x - 7$

### Image Element Undefined

$g: \R \to \R$ defined as: $\forall x \in \R: x \mapsto \dfrac 1 x$

### Image Element Multiply Defined

$h: \R \to \R$ defined as: $\forall x \in \R: x \mapsto \begin{cases} x + 1 & : x \ge 0 \\ 0 & : x \le 0 \end{cases}$

### Mapping not Well-Defined

$\theta: \Q \to \Z$ defined as: $\forall m, n \in \Z, n \ne 0: \dfrac m n \mapsto m + n$

## Quotes

Although there are revolutions in mathematics, they are usually changes in viewpoint or philosophy; earlier results do not change -- although the hypotheses needed to prove them may. In fact, there is a word in mathematics for previous results that are later changed; they are called 'mistakes'.
--2004: Ian Stewart: Galois Theory (3rd ed.): Historical Introduction.

## Linguistic Note

The word howler as a synonym for mistake derives from the tendency of those encountering it to howl with laughter in ridicule of the person committing it.

The word bug originates from the early days of the computer industry, where a malfunction was found to be caused by an insect, or bug in colloquial American, whose intrusion had compromised the workings.