@johncarlosbaez Speaking as a math educator who believes that a major barrier in math education is the way we mathematicians abuse and muddle notation in the name of convenience (cf. A/B as “A divided by B” or “A mod(quotient structure sense) B”, “field extension of A” among other disparate usages), I really don’t think we have a leg to stand on here.
@bertwells - quotienting is a categorified form of division, so we now see the ambiguity you mention as a case of our notation being smarter than we are - just like using × both for the product of numbers and the product of sets.
@johncarlosbaez Yes, of course. I especially remember the feeling of wonder when I first learned of Lagrange's Theorem and how it was the first opening of the door for me to the structured relationship of groups to subgroups. Quotienting out sets is a big deal, and the notation A/B allows us a shorthand for a set of cosets that -yes- can be thought of as a form of division of sets...
...but I don't think that is a case of the notation being "smarter" than we are, whatever that may mean. Certainly, it is a clever thing that people have done with notation. But in education I have come to be wary of cleverness when it comes at the cost of ambiguity. Because students vary wildly in cleverness, and the cognitive load of reconciling a strained "division metaphor" can really put a damper on the pedagogical activity of actually teaching, for instance, how Z/nZ actually builds a set of residue classes as cosets.
Your initial point (I think) was that it is a problem if a symbol that stands for some important types of equivalence relations gets muddled up by also denoting "is approximately equal to". I guess my own point is that humans seem to always be looking for ways to muddle and stretch the linguistic metaphors, referents, and notations that we use. Math benefits from this activity sometimes, because sometimes we land on a clever way to re-use notation and extend an idea into a new area. But there is always a cost to this activity, in ambiguity.
@johncarlosbaez There are many mistakes in the Unicode names and even obvious spelling mistakes. However, Unicode is immutable and by agreement past mistakes will never be changed.
"There are many mistakes in the Unicode names and even obvious spelling mistakes. However, Unicode is immutable and by agreement past mistakes will never be changed."
You just need another mapping. You already have the mapping from the character to the Unicode name. Now you need another mapping from the Unicode name to the proper semantics. Then you compose the two mappings to get the mapping you want.
(We software people are used to these sort of bodges.)
"~" also means "is distributed as" with a random variable on the LHS and a distribution family on the RHS.
Unicode explanations are usually written by character set experts, whose grasp of mathematics is mostly that they think it's a pain in their rear.
(But, confession: I also in the past used "~" as "approximately equal". Because I often had to type it into an email where non-ASCII was no end of pain to type, and absolutely no guarantee of what would come out on the recipient's end.)
@johncarlosbaez
I was starting to say "They do this on purpose just to irritate you, you know."
But wait, last time I looked, this is in the Unicode spec itself.
So either this character is used in the real world in a way you disapprove of (so it's in Unicode's sources that the standard is based on), OR, Unicode has made a mistake and needs to be corrected.
The latter is possible, but you'd need to be sure of other mathematician's usage, which tends to be difficult to ascertain without looking at a lot of the literature.
@johncarlosbaez Sadly those descriptions are part of the official Unicode standard. There are several others that are just mistakes and we're stuck with them.
@uffe1974 - other comments have said you're right. It's wrong within mathematics, but I probably won't have the energy to argue with the Unicode consortium.
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