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I was reading this text about the new math movement, there's a line in which he says:

Easy as it looked, teachers didn't always get the notion of "set" straight themselves, and could teach the most egregious confusions as truth. One textbook lesson plan suggested that the teacher, as an example, distinguish the subset "boys" from the subset "girls" (in the set "this class") by asking the boys to stand, and then the girls to stand, and so on; one teacher I heard about then asked "the set of boys" to stand up. But while boys, being human, can stand, sets cannot.

What's the problem of using this stand-up analogy to demonstrate the concept of set?

amWhy
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Red Banana
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    The teacher is making the mistake by commanding a set to do something, while assuming it means that the contents of the set should do that. In some sense it is failing to distinguish between $f:X\to Y$ and the related function $f:P(X)\to P(Y)$ given by $f(S)={y\in Y:\exists x\in S,(y= f(x))}$. – yearning4pi Mar 04 '13 at 05:12
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    In ordinary language, the distinction between what a set does and what the members of a set do is not so clear. It is quite common in English to say "the class stood up" when what you mean is "the members of the class stood up". – Robert Israel Mar 04 '13 at 05:29
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    The set ${2,4,6,8}$ isn't even. Same thing. – Robert Mastragostino Mar 04 '13 at 05:37
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    The command "Platoon, halt!" is often obeyed as if what were actually said was "Every soldier in the platoon shall halt!" – coffeemath Mar 04 '13 at 06:35
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    @coffeemath: And if it isn’t, someone’s in trouble. (I’m now trying to imagine my drill instructors of forty-some years ago bellowing ‘Company, collectively, halt!’.) – Brian M. Scott Mar 04 '13 at 09:03
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    Ambiguity arises when a property applies to both the set and its elements, e.g. $,{0,1}$ is even (as an ordinal), but its elements are not all even. Similarly for $,{ {}}$ is nonempty, $,{\Bbb N, \Bbb Z}$ is finite, etc. – Math Gems Mar 04 '13 at 21:33
  • @RobertMastragostino The set ${2,4,6,8}$ is even as an ordinal, as are its elements, so there is no ambiguity there. But ${0,1}$ is even and its elements are not even, so that is problematic. See my prior comment. – Math Gems Mar 04 '13 at 21:43
  • It's a fallacy of composition. If you stand up, is that the same as each one of your cells standing up? – Neal Mar 05 '13 at 13:15

2 Answers2

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I think the point being made is this: boys are elements of the "set of boys". Asking the set of boys ("set" is singular) to "stand up" is not equivalent to asking the elements of the set (boys) to stand up. In this case, the elements (boys) of the set can stand up, walk, talk..., but the set of boys is, well, a set of boys: it contains the boys in the class.

It would be perfectly appropriate to ask the elements of the set of boys to stand up. Or better yet, to ask "if you are an element belonging to the set of boys, please stand," or even "if you're a member of the set of boys, please stand up." Those are different requests than asking the set of boys to stand up.

As aptly pointed out in comments above: In natural language, people often don't distinguish between a set and its contents. After Valentine's Day, e.g., I likely reported that "I ate an entire box of chocolates in two days!", and I certainly would not have meant that I actually ate the box containing the chocolates! So "real life examples" of the failure to distinguish between a set and its contents occur very frequently, and in real-life, making such a distinction can actually sound awkward, indeed!

I think in the example you quote, the author is trying to argue that "how sets are being taught in school" may be problematic, and in the case at hand, given that the request was made as part of an educational lesson about sets, subsets, and so on, the failure to distinguish between the "set of boys" and the "boys contained in the set" is seen as problematic, pedagogically speaking.

amWhy
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This may not help as an answer (was too wordy for a comment) but I disagree that this is a bad example of sets in the following sense.

Take $i$ boys and $j$ girls. The class in the quote is represented by $|C|=i + j$. Then looking at the set $C=\{\underbrace{x_1, x_2, ..., x_i}_{\text{Boys}}, \underbrace{y_1,y_2,...,y_j}_{\text{Girls}}\}$.

If we then take the set of boys $B\subset C$, and the set of girls $G \subset C$, where $B=\{x_1,x_2,...,x_i\}$ and $G=\{y_1,y_2,...,y_j\}$, then clearly $B\cup G$ makes up all of the elements of $C$.

To me there are definitely worse ways to explain sets to people who don't understand them. The only reason that this may be confusing is that, as others have suggested, the teacher 'implies' that elements do something. But I think that this is still valid.

MITjanitor
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    You’ve missed the point, I’m afraid, and this answer doesn’t really address the question. There is nothing wrong with using the subset of boys and the subset of girls as examples of subsets of the class; that isn’t the objection. The point is that (in your notation) it is nonsense to tell $B$ to stand up: $B$ is a set, not something capable of standing up. The teacher should have told the elements of $B$ to stand up. – Brian M. Scott Mar 04 '13 at 08:57