Timeline for A question of notation: what does $x\lt y\in S$ mean?
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Jun 30, 2023 at 13:43 | comment | added | LSpice | Re, I have waged a quixotic battle against that all my years of teaching. I use $\sin(x)^2$, which students insist on reading as $\sin(x^2)$ because parentheses seem to mean nothing to them, so I am stuck with the abomination $\bigl(\sin(x)\bigr)^2$, which makes no-one happy. | |
| Jun 30, 2023 at 5:33 | comment | added | François G. Dorais | Actually, there's more like the annoying $\sin^2(x)$ convention. | |
| Jun 30, 2023 at 5:24 | comment | added | François G. Dorais | The only "universally accepted ambiguity" I can think of is precedence of operations. Everywhere, even outside mathematics, $ab+c$ means $(ab)+c$ and not $a(b+c)$. | |
| Jun 30, 2023 at 5:21 | comment | added | François G. Dorais | By "ambiguous" I mean something that has two or more plausible meanings even with full context. For example, every paper or book on partial combinatory algebras says somewhere that $abc$ means $(ab)c$. That's because the application operation is not associative and thus $abc$ is ambiguous. Interestingly, this is done even though everybody in the field uses this convention. | |
| Jun 30, 2023 at 5:15 | comment | added | François G. Dorais | Ironically, I think we're arguing about the meaning of ambiguity, not about mathematical practice. What you describe is what I would call imprecise or inexact. These are cases where the meaning can be clarified with additional context. When you say "the group $G$" it is (unambiguously) clear that you mean that $G$ is endowed with a group structure that satisfies the group axioms. It is not immediately clear what this structure is but that just missing context. I think this is part of language and it has nothing to do with mathematical practice per se. | |
| Jun 30, 2023 at 3:49 | comment | added | LSpice | Re, also, I think that that analogy may not be perfect; a pen is a pen regardless of its colour, but a set is simply not a group, even if it has a canonical group structure (sometimes a unique one up to choice of identity, but never, except the one-point set, genuinely a unique one). | |
| Jun 30, 2023 at 3:48 | comment | added | LSpice | Re, I think that "the monoid $\mathbb Z$" has at least two plausible readings, and that it is quite common to omit the operation if the context makes it clear (though also quite common to specify it). (For that matter, it is possible, though not likely, to put different group operations on $\mathbb Z$, so even "the group $\mathbb Z$" is technically ambiguous.) But I was not claiming that ambiguity is good, or even acceptable, in this particular case, only that I think that the dictum "never write ambiguously" is not commonly practiced. | |
| Jun 30, 2023 at 2:56 | comment | added | François G. Dorais | @LSpice Ambiguous usually means something that has two or more plausible readings. My personal favorite is "$1=0!$" I don't think "the group $G$" is ambiguous in that sense. Nobody would say the phrase "this is a pen" is ambiguous just because you don't know the color. | |
| Jun 30, 2023 at 1:40 | comment | added | LSpice | Nonetheless, there are surely universally accepted ambiguities. Not only does (almost?) no-one refer to "the group $(G, \cdot)$", and almost everyone to "the group $G$" (in general; of course one might write something like "the monoid $(\mathbb Z, \cdot)$ versus the monoid $(\mathbb Z, +)$"), doing so would surely be regarded as obscuring, not clarifying. So I think one cannot respond: always write precisely!; but rather: only use almost universally understood ambiguities! And from this point of view the question can be understood as: is this amiguity universally understood, and, if so, how? | |
| Jan 7, 2021 at 17:00 | history | migrated | to mathoverflow.net | ||
| Dec 23, 2020 at 4:41 | history | answered | François G. DoraisMod | CC BY-SA 4.0 |