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    $\begingroup$ Creating a new thread is unnecessary and bad idea, e.g. see here and its links $\endgroup$
    – Bill Dubuque
    Jan 6, 2022 at 10:42
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    $\begingroup$ I applaud annual lists, lest the lists of "answers" becomes difficult to navigate. Also, @Paramanand, thanks for including the links to other "volumes", previously! $\endgroup$
    – amWhy
    Jan 6, 2022 at 20:31
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    $\begingroup$ I'm sorry to be nitpicky about this, but the second bullet point may need some further tweaking. Rather than saying, "Notice that the first edit after the question may push the question into the reopen review queue if the edit is done within 5 days of closure...", it might be better to just say "Any question with significant edits that address the close reason(s) are automatically put into a review queue to be considered for reopening", which is the text in the corresponding Help Center article. $\endgroup$
    – The Amplitwist
    Jan 22, 2022 at 9:57
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    $\begingroup$ @BillDubuque My answer here that you said should not be deleted, was just deleted. math.meta.stackexchange.com/questions/34447/… $\endgroup$
    – Nick Alger
    Jun 20, 2022 at 22:41
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    $\begingroup$ @Nick Flag it for moderator attention. $\endgroup$
    – Bill Dubuque
    Jun 21, 2022 at 2:01
  • $\begingroup$ @Bill, the answer on meta has now been deleted and locked by "Community" so it can no longer be flagged. $\endgroup$
    – Gerry Myerson
    Jun 26, 2022 at 0:16
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    $\begingroup$ @Gerry the deleted answer is still flaggable for me. In any case one can raise the flag anywhere (e.g. on this question), and Nick has already done so (cf. Math Mods' Office chat). $\endgroup$
    – Bill Dubuque
    Jun 26, 2022 at 3:11
  • $\begingroup$ Why is one user allowed to dominate this thread? If you look at the posts, and the requester who always gets their way, often after inappropriate edits to a poor question, there's a definite pattern of dominance. And I say this only because mods have not yet intervened. I'm surprised that the moderators keep turning their heads in silence and inaction, hence, giving free rein to one user. $\endgroup$
    – amWhy
    Nov 10, 2022 at 21:14
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    $\begingroup$ @amW, anyone is allowed to post to this thread, and anyone is allowed to register disagreement with a requester. If other users disagreed with any one requester's posts, they could vote that user down, with no need for moderator intervention. And if other users haven't complained about one particular user, then I'd say the moderators are quite right not to intervene. $\endgroup$
    – Gerry Myerson
    Nov 11, 2022 at 5:35
  • $\begingroup$ @GerryMyerson Not practically. You play a huge role in that. $\endgroup$
    – amWhy
    Nov 18, 2022 at 20:41
  • $\begingroup$ Not exactly a welcoming interaction here. Unbelievable. math.stackexchange.com/questions/4616434/… $\endgroup$
    – Alper
    Jan 14 at 18:14
  • $\begingroup$ @Alper I just cast the last needed undelete vote to that posting, so now the posting has been undeleted. However, just so you know, the posting is still closed. I also cast a re-open vote for the posting. $\endgroup$
    – user2661923
    Mar 4 at 23:43
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    $\begingroup$ @user2661923 that post was redeleted a couple of weeks ago. $\endgroup$
    – Gerry Myerson
    Mar 28 at 2:41

66 Answers 66

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Undeleted by users then deleted by bot.

Please undelete and reopen this:

https://math.stackexchange.com/q/4618260/104041

The question is mine.

I put an awful lot of effort into it. The downvotes confuse me and I have no clue why it was deleted (beyond the automatic deletion when there's enough downvotes with no answer).

The question is relevant to my research and it's important to have it easily accessible.

I believe it to be high quality.

The main feedback objections seems to be a matter of style, not substance: the notation used and the layout of the thing. It reads better on a mobile than on the main site, I believe; but what does it matter?

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    $\begingroup$ FYI this question was never closed, so there's no need to ask for a reopening. And the deletion was automatic, as per this criteria. $\endgroup$
    – KReiser
    Nov 25 at 18:25
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I answered this question around 3 years ago.

7 months ago, it was commented, in bold no less, that there was an error. This brought my answer from a +4 score down to +2. I reviewed the comment and the supposed error did not feature in my answer. I explained as much in a comment back. This morning I woke up and my answer score is -1 and my answer has been deleted. Another comment was left yesterday, this time decrying my comment as handwaving and claiming that I'm making the same error (to reiterate: I'm not) and showing it doesn't work with a counterexample against the wording of the question (which is specifically about a set of consecutive integers, like my answer).

I've since edited my answer to flesh out the details a bit more and really spell it out, but it feels very unfair to have someone declare and insist upon an error without really trying to understand my answer.

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    $\begingroup$ I'm just reading your answer, and the first issue I have is that there is no quantifier attached to the $a$ in the first bit of working. I presume it is an $\exists$, and comes after the initial $\forall$? I'm then unsure precisely what the $k$ is, but I don't think we can "match" $1\cdot2\cdots n$ with the sequence $(k+1)(k+2)\cdots(k+n)$, because we have no control over the $a$ - unless the $a$ is actually quantified by a $\forall$. $\endgroup$
    – user1729
    May 2 at 8:37
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    $\begingroup$ Another problem, which I think is the one BillDubuque is pointing out, is that your answer seems to only really rephrase the fact that "one in every $t$ consecutive integers is divisible by $t$." This is true, but more is needed here, as you will be double-counting these sequences. So you need to flesh out the claim "thus the same is true across the whole product". $\endgroup$
    – user1729
    May 2 at 8:48
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    $\begingroup$ (For example, for $4!$ we need sequences of $4$, $3$ and $2$ integers in our product, and without further argument these sequences need to be disjoint, i.e. we need to rewrite $4!$ as the product of $4+3+2=9$ consecutive integers, but this is impossible!) $\endgroup$
    – user1729
    May 2 at 8:50
  • $\begingroup$ I agree, $a$ ought to have it's own $\exists$. $a$ is decided by $k$ and $n$. If we are multiplying, as an example, $123\cdot \ldots \cdot 128$, then $n=6$ as there are six numbers, we pick $a=21$ as $21\cdot 6=126$ is in our set. Then our numbers go from $123\equiv 3 \pmod 6$ to $128\equiv 2 \pmod 6$. $\endgroup$
    – Rhys Hughes
    May 2 at 9:07
  • $\begingroup$ Regarding the second problem, I can see the concern, however it works out with the same observation that we are cycling the residues $\mod n$. I fleshed out the details in what I added earlier, but in essence, if $n$ has a divisor $d$, there will be a number $x$ in our set with $x\equiv d\pmod n$. Take the same set I used earlier, we have $6=2\cdot 3$ and $123\equiv 3\pmod 6$ while $128=2\pmod 6$. $\endgroup$
    – Rhys Hughes
    May 2 at 9:15
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    $\begingroup$ Instead of fleshing out the details below, you should edit your answer so that it is clearer in a first reading. $\endgroup$
    – user1729
    May 2 at 14:02
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    $\begingroup$ The argument is either incorrect or seriously incomplete. It is impossible to decide which since you have not supplied rigorous justification of the huge gap, i.e. precisely how the OP's claim is inferred from "one in every $t$ consecutive integers is divisible by $t$". I doubt that anyone has any idea how you propose to do so based on what you have written. $\endgroup$
    – Bill Dubuque
    May 2 at 18:05
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Please consider reopening Proving that $\langle x, y \rangle$ is a not principal ideal of $\mathbb{Q}[x, y]$

Although related, the current linked question, which is closed too, asking for an example, is not a duplicate. Moreover, the post is linked to a chain of questions that are all closed.

If this question is to be closed, there is supposed to be a better "duplicate" target: in the all closed chain of questions, NONE new answer is allowed to added.

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  • $\begingroup$ This proof in the linked dupe does exactly what the OP seeks (put $c = y\in\Bbb Q[y]$) so it is not clear what you mean by "better dupe target". There are many possible dupe targets (chase the links) for this common exercise, and nothing at all novel in the (duplicate) answers posted. [Your profile shows zero activity in abstract algebra. Experience in a field is usually necessary in order to properly assess duplicates]. $\endgroup$
    – Bill Dubuque
    Aug 24, 2022 at 1:38
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    $\begingroup$ It seems that you have a very different definition of "dupe" from mine, whatever it is. Experience in a field $\Large{\neq}$ activities in your mathSE profile. "better dupe" means one does not need to "chase the links", not to mention that it is linked to a closed question that may be voted to be deleted by users. $\endgroup$
    – user1046533
    Aug 24, 2022 at 1:41
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    $\begingroup$ In my experience it is - esp. for most users whose tags don't go much beyond calculus. Please explain what you mean by a "better dupe target". We can easily add more dupe links. $\endgroup$
    – Bill Dubuque
    Aug 24, 2022 at 1:43
  • $\begingroup$ Yes, usually, maybe. $\endgroup$
    – user1046533
    Aug 24, 2022 at 1:47
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    $\begingroup$ Your recent comments (and votes) on main made it clear that you are strongly against abstract dupes. But this is only a very slight abstraction - viz. specializing the nonzero nonunit $c$ to $y\in \Bbb Q[y]\,$ in the Lemma in the dupe. It would be greatly obfuscatory to have separate questions for various choices of nonunits $c$ in motley domains. A primary goal of abstract algebra is to prove theorems at their natural level of abstraction - so they apply as widely as possible. $\endgroup$
    – Bill Dubuque
    Aug 24, 2022 at 2:02
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    $\begingroup$ "Your recent comments (and votes) on main made it clear that you are strongly against abstract dupes." Please don't put labels on me Bill. I do not want to discuss anything in general under this thread.// Comments are not supposed to be answers in main; one should address the issue Arturo mentioned in meta if one wants to handle abstract dupe: (cont.) $\endgroup$
    – user1046533
    Aug 24, 2022 at 2:27
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    $\begingroup$ (cont.) *It would require the "survivor variant" to have answers that address both the general strategy (so as to be useful for future minor variants) and specific application to the question at hand (which shows how the general strategy applies to a specific problem). * $\endgroup$
    – user1046533
    Aug 24, 2022 at 2:30
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    $\begingroup$ This is probably a more exact dupe target (containing the "survivor variant"). $\endgroup$
    – Sarvesh Ravichandran Iyer
    Aug 24, 2022 at 10:08
  • $\begingroup$ @Sarves But that is already true for the existing linked dupe (case $K = \Bbb Q$; the simple proof works for coefficient field $K$ and no simplification arises when $K = \Bbb Q$) $\endgroup$
    – Bill Dubuque
    Aug 24, 2022 at 15:35
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    $\begingroup$ @BillDubuque You're right, the same proof works with $K$ replaced by $\mathbb Q$, but I think you should leave a comment mentioning this , however obvious it might seem. Heck, I'll do it. $\endgroup$
    – Sarvesh Ravichandran Iyer
    Aug 24, 2022 at 15:51
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    $\begingroup$ @Sarvesh Any student studying ideal theory in rings that doesn't see immediately how to specialize a theorem on fields to the field $\Bbb Q$ has seriously put the cart before the horse. This goes back to my original comment - it is crucial for those assessing dupes to have significant experience in the topic in order to understand which inferences can be assumed trivial and left to the reader. $\endgroup$
    – Bill Dubuque
    Aug 24, 2022 at 16:25
  • $\begingroup$ This post has been deleted. $\endgroup$
    – The Amplitwist
    Nov 20, 2022 at 13:35
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It's inconsistent to close How to become good in Mathematics?, but keep open How do I get good at Math? open.

Please reopen the former, and flag it as duplicate of the latter?

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    $\begingroup$ I do not see what benefit opening this question would give. Both questions are very old, and would be closed nowadays. The first question was asked when the site was still quite young, which is why it was likely left open. $\endgroup$
    – user1729
    Oct 2 at 11:25
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Please reopen Introductory Group theory textbook.

It's illogical for this to be closed as "opinion-based", when introductory textbook recommendations remain open.

Books on Number Theory for Layman
Good Book On Combinatorics
What is the best book to learn probability?

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  • $\begingroup$ The three questions you link were all asked twelve years ago, which is like four generations on math.stackexchange. And you've now made them targets for closure.... $\endgroup$
    – Gerry Myerson
    Jun 2 at 6:48
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    $\begingroup$ Now I've noticed the Group Theory one is also twelve years old (and it was posted by a fellow who was enrolled in my Group Theory course at the time!). $\endgroup$
    – Gerry Myerson
    Jun 2 at 6:55
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    $\begingroup$ While I do think book recommendation questions have a place at MSE, but I am unsure about this specific one. The highest-voted answer is about Rotman's book, and I do not think this is a good answer (of course, this is my opinion - see the close reason! - but it does have some extremely complicated content, and for example the pretty standard "classification of f.g. abelian groups" comes after less-standard stuff on Classical and Mathieu groups). Moreover, this question is not going anywhere - it is just closed rather than deleted, and questions can still be closed as a duplicate of it. $\endgroup$
    – user1729
    Jun 2 at 11:26
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Please reopen Does ‘that is’ (‘i.e.’) mean ‘equivalently’ or ‘so’ or depend on the context?, whose tag, article-writing, goes: “Various aspects of writing mathematics such as style, notation, grammar, frequently used phrases and common mistakes”.

When re-opened, I will post this Answer.

Regarding the phrase ‘that is’/‘that is to say’ in the sense of the abbreviation ‘i.e.’, although some of its definitions seem to suggest equivalence, in practice, its logical flow actually runs the gamut:

  1. $\boldsymbol\iff$  (in other words, clarification)

    • Triangle A is equilateral; that is, its sides are of equal length.

    • The sinks of homes with hard water—i.e., a high mineral content—will gradually accumulate white or yellow buildup.

  2. $\boldsymbol\implies$  (elaboration, in other words)

    • Figure A is equilateral triangle; that is, its sides are of equal length.

    • You’ll love how your face feels with the new Guide-n-Glide razor; i.e., it will feel handsome, fresh, and clean.

  3. $\boldsymbol\Longleftarrow$  (clarification, elaboration)

    • One of the intercepts of $(x-3)^2,$ that is, its $x$-intercept, is a minimum point.

    • One meal, i.e., breakfast, is included in the price of the room.

While it is understandable that careful writers wonder whether a sentence like ‘if $P(x),$ then $-2<x<3,$ that is, $|x|<3$’ is technically correct or whether it is peculiar to mathematical writing, the above examples demonstrate that the phrase ‘that is’ (‘i.e.’) has no niche usage that is special to mathematics/technical communication.

In case this phrase's logical meaning is not disambiguated by the context and its ambiguity is not inconsequential, it's worth noting that only Meanings 2 & 3 can unintentionally convey an unintended point.

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    $\begingroup$ The use of "i.e." in math is no different from its use elsewhere in English. It is not a math question. $\endgroup$
    – Xander Henderson Mod
    Sep 28 at 3:16
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    $\begingroup$ @XanderHenderson It is a math question and your first sentence is a possible answer to it. $\endgroup$
    – Hayatsu
    Sep 28 at 4:19
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    $\begingroup$ There is no reason to assume that the use of "i.e." in maths is different from elsewhere, and the question does not give any evidence towards this. If the answer was "it is different for these reasons", or if the question was about how to read a specific sentence which didn't parse with the usual use of i.e. then the question would be reasonable. But as-stated, it could be asking about the use of any random word, like "it" or "these" or "hence", and I cannot see what benefit it brings. $\endgroup$
    – user1729
    Sep 28 at 8:53
  • $\begingroup$ @user1729 A quibble just with your first sentence: accuracy is particularly valued in mathematical writing, so it is natural for careful writers to wonder about this discrepancy: many dictionaries' explanation of 'i.e.' as indicating equivalence versus sentences like "figure A is equilateral triangle, i.e., figure A's sides are of equal length" even in mathematics. $\endgroup$
    – ryang
    Sep 28 at 9:24
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    $\begingroup$ @ryang That isn't an explanation, but an (unhelpful) mnemonic (this is wikipedia and there is no citation for the mnemonic, so it is unclear if anyone actually uses this anyway!). Having now just looked in a few dictionaries, the correct use of i.e. in common English is identical to the correct usage in mathematics, and I have seen no evidence to the contrary. Indeed, apparently the APA style guide states that all instances of "i.e." be changed to "that is". $\endgroup$
    – user1729
    Sep 28 at 10:15
  • $\begingroup$ @user1729 Eeeeps, I hadn’t noticed that my previous link was talking about a mnemonic! Anyhow, the point is that most folks will confidently declare that ‘ie’ has the sense of equivalence, which is inconsistent with how this phrase can be used in mathematics; in this view the fact that this phrase turns out to not differ in usage between eng & math doesn’t negate the query as mathematically founded; I’d many times asked myself the OP’s exact question, so I believe it is pertinent and useful to a subset of mathematics writers. $\endgroup$
    – ryang
    Sep 28 at 10:41
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    $\begingroup$ Half the readers on, say, ELL.SE having trouble distinguishing among the logical sense of my 3 above example pairs lends credence to why this discussion is not an ill fit for math.SE. Anyhow, here’s a pertinent remark from a previous discussion: “The point is that an outsider would not be able to know a priori whether the term is used just as in natural language or in a technical sense. If it had happened that the word (such as "or") is used in a technical manner, asking someone who is not a mathematician would not lead to the right answer other… $\endgroup$
    – ryang
    Sep 28 at 10:55
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    $\begingroup$ …than by accident. In this one instance it just so happens that the word does not have a separate mathematical way of being used. But, again, assessing that can essentially only be done from within the community.” $\endgroup$
    – ryang
    Sep 28 at 10:55
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    $\begingroup$ The question is not a question about mathematical English. It is a question about English, plain and simple. The use of "i.e." is no different in mathematics than it is anywhere else in the English language. Math SE is not the right place to have this question answered. Indeed, there are already many, many answers on English Language & Usage: google.com/… . $\endgroup$
    – Xander Henderson Mod
    Sep 28 at 12:19
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    $\begingroup$ @ryang I understand what you are saying, and I agree with your point that the phrase turning out "to not differ in usage between eng & math doesn’t negate the query as mathematically founded". However, my point is there is no reason to assume mathematical usage differs from the "dictionary" definition. Questions like these should be about clarifying well-founded doubt, not asked on a whim. $\endgroup$
    – user1729
    Sep 28 at 13:09
  • $\begingroup$ Does anybody know if "i.e." is more frequent in math texts than in non-math ones? I'm not sure how to do that kind of search. $\endgroup$
    – Matthew Leingang
    Sep 28 at 17:55
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    $\begingroup$ @user1729 [Regarding the substance of the OP's question, not its suitability for this site] I expanded my Answer to explicitly support (what I infer is) your position that the dictionary definition of 'i.e.' is compatible with all the three logical senses (⟺ ⟸ ⟹). $\endgroup$
    – ryang
    Sep 29 at 6:02
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