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Several languages (including Mathematics) use polysemy.

My question is why?

Specially in mathematics, where precision is important, polysemy seems to be undesirable but it seems it is inevitable in any language. Why?

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3 Answers 3

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The simple answer is that that's the way that language is.

Almost all languages are not designed: they just happen, and develop without anybody consciously shaping them.

Humans are tolerant of a great deal of ambiguity (of which polysemy is one instance). If a particular ambiguity in a language becomes troublesome, then speakers will find ways to disambiguate as far as is necessary.

An example is when reduction of unstressed final syllables make different grammatical cases fall together, as in Middle English and many other languages: the whole mechanism of case marking disappears, and the language finds other ways to express the relationships formerly marked by case endings. In a sense, the words are now polysemous, in that they are covering semantic fields formerly covered by distinct words - the word with different inflections.

Polysemy, like everything else in human language, arises because people start using it, and it doesn't impede communication (or, if it does, something else changes to restore lost facilities).

Edit: as for mathematics: as others have pointed out, mathematics isn't a language, so I guess you're talking about mathematical notation. Some symbols are used for the different purposes in different specialities, which does not normally cause any confusion. In rare cases where there might be confusion, the notation is altered. For example, I believe that electrical engineers use j for the square root of -1, to avoid confusion with i = current.

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  • Even if mathematics isn't considered a language, mathematical English (see, for instance, abstractmath.org/MM/MMMathEnglish.htm) can be thought of as special form of English.
    – J W
    Commented Dec 2, 2022 at 18:34
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Languages aren't monoliths, they are spoken by many individuals, and every individual uses the language somewhat differently (speaking an ideolect, as it is called). The individual differences introduce polysemy in a natural way, and humans can cope with it, so there is no need to weed it out in natural languages.

I cannot speak about mathematics here, but mathematics isn't a language anyway.

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  • whw do you think mathemstics is not a langaguage? Commented Dec 3, 2022 at 14:30
  • The space in a comment is too short to explain it in detail, but mathematics cannot express everything a human language can express. Commented Dec 3, 2022 at 20:17
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Along the lines argued by Colin Fine, the main reason is that absolute precision and objectivity in definitions is impossible. To convince yourself of this, pick 12 words (as non-random as you want), and attempt to state the meaning of the terms using only existing absolutely precise objective definitions. You should at least include the words "dog" and "eat" in your set.

The defining properties of any word are somewhat flexible and subjective, and highly contextual (is a dingo a dog? is a tomato a fruit?). There are marginal cases for many words, for instance is Afrotyphlops schlegelii a worm (no, but many people think it is) and a penguin a bird (technically yes, but not everybody reads science books telling them this technical fact). Polysemy frequently arises when a word is used in a novel context, carrying over some but not all defining properties of the earlier definition and applying it to a novel context. This is how "dog" came to also be a verb, likewise "stoned" arose by metaphorically applying some of the defining characteristics of a kind of mineral to a person.

Seemingly-precise scientific terminology can become very imprecise by this mechanism of dropping part of the definition and substituting new criteria, while retaining the label. One example in linguistics is "marked", which originally was the metaphorical application of a graphic concept to grammatical analysis to express one theoretical idea (Trubetzkoy's), but then the term was retained and the definition reworked again and again, thus "marked" is highly polysemous in linguistics. The term "C-command" was originally coined to refer to a specific kind of structural relation in syntax (a specialization of earlier more general "command"), then as it became popular, the term was retained and the definition was changed every few years. For mathematics, you might want to look at this article, taking for example the multiple definitions of "neighborhood" in mathematics.

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