fix

In mathematical statements, mathematical objects such as points and numbers are described as being fixed. A possible meaning for this usage is that the mathematical object in question is not allowed to vary throughout the statement or proof (or, in some cases, a portion thereof). Although a fixed object typically does not vary, it is almost always arbitrary. This may seem paradoxical, but it is quite logical: An object is chosen arbitrarily, then it is never allowed to vary. See the entry betweenness in rays for an example of this usage.

The usage of the fix and fixed may also that a mapping sends the mathematical object to itself. These two usages are technically not the same. The former usage (described in the previous paragraph) states a property of the mathematical object in question and is always either part of an implication (as in “If $x\in ℝ$ is fixed, then…”) or a command made by the author to the reader (as in “Let $x\in ℝ$ be fixed.” and “Fix $x\in ℝ$.”). The latter usage (described in this paragraph) states a property of a mapping and may or may not be part of a conditional statement or a command. The word “fixes” always refers to this usage (as in “Note that $f$ fixes $x$.”). See the entry fix (transformation actions) (http://planetmath.org/Fixed) for a further explanation of the latter usage.