fixFix22006-08-21 23:15:462006-08-23 16:34:42Definition\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{psfrag}
\usepackage{graphicx}
\usepackage{amsthm}
\usepackage{xypic}
\PMlinkescapeword{objects}
\PMlinkescapeword{object}
\PMlinkescapeword{states}
\PMlinkescapeword{word}
In mathematical statements, mathematical objects such as points and numbers are described as being \emph{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 \PMlinkescapetext{words} \emph{fix} and \emph{fixed} may also \PMlinkescapetext{mean} 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 $x \in \mathbb{R}$ is fixed, then...'') or a command made by the author to the reader (as in ``Let $x \in \mathbb{R}$ be fixed.'' and ``Fix $x \in \mathbb{R}$.''). 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'' \emph{always} refers to this usage (as in ``Note that $f$ fixes $x$.''). See the entry \PMlinkname{fix (transformation actions)}{Fixed} for a further explanation of the latter usage.