Why do we use polysemy?

Question

From Polysemy - Wikipedia:

Polysemy (/pəˈlɪsɪmi/ or /ˈpɒlɪˌsiːmi/; from Ancient Greek πολύ- (polý-) 'many', and σῆμα (sêma) 'sign') is the capacity for a sign (e.g. a symbol, a morpheme, a word, or a phrase) to have multiple related meanings. For example, a word can have several word senses. Polysemy is distinct from monosemy, where a word has a single meaning.

Several languages (including Mathematics) use this kind of linguistic resource. My question is, "Why?" Specially in mathematics where precision is welcome polysemy seems to be undesirable but it seems it is inevitable in any language. Why?

Answer 1

I believe there can be a clash between the ideal local symbol and a more global symbol that clashes with the local symbol.

When discussing a theorem concerning polyhedra, I usually use $P$. But then later I might need two polyhedra. This could be $P$ and $P'$, especially if $P'$ is directly related to $P$, say, $P'$ is a truncation of $P$. But continuing the exposition, I may need a 3rd polyhedron, but $P''$ starts to become ugly, and maybe I should go back and set $P=P_0$, $P_1$, $P_2$.

But suppose then I need to discuss the unfolding of $P$ to a polygon in the plane. Normally, I would want to use $P$ for a planar polygon, but here I might opt to call the polyhedron $\mathcal{P}$ and the polygonal net $P$.

So I find when writing that the ideal local symbol is often evident, but as the context widens, adjustments need to be made to respect naturalness but include more complex environments.

At an extreme, one would never want to exclude $d$ to represent disatance just because later on one needs to refer to $dx/dy$. The context easily distinguishes the two meanings of $d$, and to refer to distance $u$ or $\rho$ would only obscure.