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Metonymy is a figure of speech where a word or another expression is used to mean something other than its literal meaning. This phenomenon is not restricted to the "usual human languages" (such as English or Finnish) but also appears in the symbolic language of mathematics. I am not interested in drawing boundaries between English and mathematics in a mathematical presentation; I am interested in metonymy in all mathematical contexts.

Examples of metonyms:

  • When we discuss a function $f$, we often actually write or say $f(x)$, although $f(x)$ is the value of the function $f$ at $x$. That is, the value of the function at a "generic point" is used as a metonym for the function itself. (An extended example of this metonymy is the statement "The Fourier transform of $f(x)$ is $\hat f(\xi)$".)
  • A continuous function $\gamma:I\to\mathbb R^n$ from a real interval $I$ is a curve. "The curve $\gamma$" occasionally also refers to the image $\gamma(I)$, the trace of the curve, by metonymy. Different authors use different conventions here, but the word "curve" is used in various different meanings.
  • A group is a set together with a (certain kind of) binary operation. The set alone is often used to denote the group by metonymy, the operation being understood implicitly.

Metonyms are always about using the same expression (word, symbol…) in several different meanings. Metonymy is often convenient, but it can also be a source of confusion if it is not introduced explicitly. Sloppy notation and abuse of notation are typically metonymous.

Questions:

  1. How should I take metonymy into account when teaching mathematics? Should I try to explicitly state whenever this linguistic phenomenon is present? In my experience this is rarely done, and I have a feeling that some students have difficulties because they do not recognize when an expression is used metonymously.
  2. What are the most important metonyms that I should make my students aware of? (I am thinking about university level education, but answers on any level are welcome.)
  3. Are there some metonyms that I should avoid or instruct my students to avoid? Do students commonly have misconceptions of a metonymous origin? This is the kind of problem that I had in mind: If a banana costs $3\$$ and an apple $2\$$, a student may be tempted to write $\text{banana}=3\$$ and $\text{apple}=2\$$ and conclude $2\times\text{banana}=3\times\text{apple}$ which is not true. This is because of using the symbol "$=$" metonymously, not actually referring to two things being equal. If one introduces a price function $p$ and lets $p(\text{apple})=2\$$ and so on, then $p(2\times\text{banana})=p(3\times\text{apple})$ is a reasonable statement.
  4. To what extent should I delay exposition to metonyms? Metonymy is a part of standard mathematical communication, and mathematical educations should prepare students for it. But this does not necessarily mean that I should express myself metonymously from day one when teaching a new topic.
  5. Can metonymy be used as a tool to help learn mathematics? That is, are there useful metonyms that I should introduce?
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    $\begingroup$ I found post focusing on the "$f$ or $f(x)$" example. It is relevant but different. matheducators.stackexchange.com/q/604/2074 $\endgroup$ – Joonas Ilmavirta Oct 1 '14 at 18:19
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    $\begingroup$ I guess you meant $2\times p(banana) = 3\times p(apple)$ $\endgroup$ – Jonas Gomes Oct 9 '14 at 21:10
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    $\begingroup$ @JonasGomes, the price functions is linear, so that is the same thing. I wanted to emphasize that it's not $a=b$ but $p(a)=p(b)$. $\endgroup$ – Joonas Ilmavirta Oct 10 '14 at 4:45
  • $\begingroup$ What about bulk discounts in non-competitive markets?!! (I kid, I kid.) $\endgroup$ – Vandermonde Nov 11 '15 at 4:13
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Metonymy and its relatives, metaphor, polysemy, synecdoche occur all over the place in mathematical writing, and sometimes cause students problems and sometimes don't, because those thought processes are basic to our understanding of everything.

Where mathematics comes from, by George Lakoff and Rafael E. Núñez. Basic Books, 2000.

Exploring the role of metonymy in mathematical understanding and reasoning: The concept of derivative as an example, by Michelle J. Zandieh and Jessica Knapp. The Journal of Mathematical Behavior, Volume 25, Issue 1, Pages 1-17. This is owned by (boo hiss) Elsevier who charges an exorbitant amount to get a copy, so I have not read it. I have not checked if it is on Jstor.

Your Metaphor, or My Metonymy?, by John Mason, 2010.

Metonymy and metaphor in mathematics, by Tony Phillips, 1999.

Handbook of mathematical discourse, by Charles Wells.

Images and metaphors, article in abstractmath.org.

Incomplete notation, section of an article in abstractmath.org.

Embedding, section of an article in abstractmath.org.

One of my favorite examples of this sort of thing is when a mathematician says something like "x^2-1 vanishes at x=1".

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