Vagueness\, or indeterminacy\, has several disputed explanations. One popular view is that vagueness is due to semantic indecision. If it is indeterminate whether Mary and Beth are friends\, or whether Mary is rich\, this is because there are several relations which we could mean by &lsquo\;friendship&rsquo\;\, or several properties we could mean by &lsquo\;being rich&rsquo\;\, but we have not decided among them.
\nWe could precisify our language in different ways\, each precisification corresponding to a decision. It is then definitely true that Mary and Beth are friends if it is true on all precisification\, or valuations &ndash\; it is supervaluationally true. \;Thus indeterminacy is really multiplicity\, multiplicity of possible meanings.
\nThis core idea can be developed in various ways. An opposing idea is that in some cases there is indeterminacy which cannot be so explained. There is a single relation of friendship\, for example\, but it is indeterminate whether Mary and Beth are friends because it is &lsquo\;fuzzy&rsquo\; in some way so that there is no saying what is the case. This appeal to a singular indeterminacy in the world is more appealing in some cases than others.
\nFor example\, a mountain or a cloud may seem to have fuzzy boundaries. It may seem tempting to say that this is because there is a single object which has\, indeed\, in some sense\, fuzzy boundaries\, and not that there are many overlapping objects\, each with equal right to be \;called the mountain\, so that &lsquo\;the mountain&rsquo\;\, is improper in somewhat the same way as\, say\, &lsquo\;the English king called &ldquo\;Charles&rdquo\; or &lsquo\;the odd number between four and eight&rsquo\;\, because of a multiplicity of candidates.
\nThe intention of the paper is to defend the thesis that indeterminacy is multiplicity against a challenge which appeals to examples including that of the mountain. My conclusion is that there is no reason to deny that the sort of puzzles discussed can be explained appealing only to a combination of (a) determinate multiplicity\, (b) the linguistic thesis that counting is sometimes not by strict identity and (c) supervaluationism.
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