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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?

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    $\begingroup$ Can you give an example of an objectionable polysemy in mathematics? $\endgroup$ Dec 2, 2022 at 16:17
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    $\begingroup$ Are you asking for general rules observed in the evolution of languages, or rather for arguments for/against having a notational reform in mathematics? The question "why?" is very broad. $\endgroup$ Dec 2, 2022 at 16:19
  • $\begingroup$ The posts matheducators.stackexchange.com/q/4475/376 and matheducators.stackexchange.com/q/2477/376 may also be of interest. $\endgroup$
    – J W
    Dec 2, 2022 at 18:10
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    $\begingroup$ From what I understand, the lack of this feature in Chinese, at least written Chinese, is one of the many reasons that English is hard to learn. One the other hand, Chinese words for numbers 0, 1, 2, .. , 9 are way more linguistically efficient and their naming systems for things like fish, birds etc. are far more logical than the English lexicon's adhoc cacophony of mackerel, cod, salmon, ... we could be here a while. In math, the issue is more with duplicate meaning for common notation. For example, what is $(1,2)$ ? To each their own. Freedom of speech in math I say. $\endgroup$ Dec 2, 2022 at 19:31
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    $\begingroup$ You should improve your question by putting an example of a polysemy in your question and explaining why you think it is harmful. Telling me that $2^x$ has different meanings when $x$ is an integer or not does not seem to me like it causes any harm. You also left a reply about "polygedra" which I don't understand at all. You should edit your question to improve it so later people who come across it can understand what you are asking. $\endgroup$ Dec 4, 2022 at 1:15

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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.

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