Timeline for Examples of bad notation and its consequences
Current License: CC BY-SA 4.0
54 events
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| Apr 23, 2023 at 18:41 | comment | added | Iosif Pinelis | @MoisheKohan : As for "ChatGPT's opinion", I don't think it has opinions. I said in the beginning of my post, "ChatGPT gives a few examples of bad notation used more or less long ago, including (i) [...]; (ii) [...]; (iii) [...]. Then I expressed my opinion on each of these three examples, in more nuanced and detailed ways than simply stating "I share (or don't share) this 'opinion'". What could I possibly have done wrong here? | |
| Apr 23, 2023 at 18:33 | comment | added | Iosif Pinelis | @MoisheKohan : First of all, you did not seem to answer any of my questions: (i) Why do you not think that "should be allowed here (at MO)"? (ii) What do you mean, exactly, by "AI-based"? (iii) What other tools would you forbid as well (and why)? | |
| Apr 23, 2023 at 18:21 | comment | added | Moishe Kohan | As in "ChatGPT gives...." If you happen to share ChatGPT's opinion on this, please say so directly instead of referring to ChatGPT. | |
| Apr 23, 2023 at 17:51 | comment | added | Iosif Pinelis | @MoisheKohan : But why? And do you mean, exactly, by "AI-based"? What other tools would you forbid as well? | |
| Apr 23, 2023 at 17:43 | comment | added | Moishe Kohan | I do not think AI-based answers (and questions) should be allowed here (at MO). | |
| Apr 23, 2023 at 14:42 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Apr 23, 2023 at 2:03 | comment | added | Iosif Pinelis | Previous comment further continued: So, I cannot see any compelling reason for us to have our own grammar here. | |
| Apr 23, 2023 at 2:03 | comment | added | Iosif Pinelis | Previous comment continued: In our case, according to the standard use of the quantifier "for all", its object is a symbol that should immediately follow it, as in $\forall x$. So, what symbol is the object of the "for all" in "for all $1\le k\le n$"? It can only be $1$, which leads to nonsense. The statement $1\le k\le n$ is a statement, and not a symbol, and therefore this statement cannot normally be the object of this quantifier. Moreover, we have good, fully grammatical alternatives, including "for all $k\in[n]$" and "for all $k\in\{1,\dots,n\}$". | |
| Apr 23, 2023 at 2:02 | comment | added | Iosif Pinelis | @Carl-FredrikNybergBrodda : You wrote: "I would read ["for all $1\le k\le n$"] aloud, and let its grammar be, as "for all $k$ between $1$ and $n$". -- I hope we can agree that, if we write a mathematical text in (say) English, then we should generally follow the rules of English grammar. Perhaps we can also agree that, whenever we choose to deviate from those rules and "let [our own] grammar be [something else]" (in your words), we should have a good, compelling reason for that. | |
| Apr 23, 2023 at 1:35 | comment | added | Iosif Pinelis | @Carl-FredrikNybergBrodda : You have been given at least three explicit opportunities to deal with your counterfactual comment "It seems the irony that the generated list is numbered with Roman numerals is lost on ChatGPT". However, you have not taken any one of these opportunities. I don't know why you chose to do so, but at this point I just want this fact to be recorded. | |
| Apr 23, 2023 at 1:32 | comment | added | Iosif Pinelis | @bob : I don't know what you mean by a knowledge base, and I never said that it can be used as such. Whether it is intelligent or not is a matter of how intelligence is defined, and this is quite irrelevant here. What I said was that it is a tool, which could be useful if used appropriately. In the present case, it immediately gave 4 or 5 suggestions, of which at least 2 were good -- not bad at all. But nobody forces you to use it! | |
| Apr 22, 2023 at 3:16 | comment | added | bob | I’m not sure you can make ChatGPT useful as a knowledge base—that’s just not what it is. It’s expert at modeling textual communication and was trained on a massive and diverse corpus, but anytime it gets a fact right that’s a happy accident. It has no concept of facts, just symbols and how they’re statistically strung together by humans. We’re the intelligent ones. It’s not. | |
| Apr 21, 2023 at 13:57 | comment | added | Iosif Pinelis | @Carl-FredrikNybergBrodda : Again, let us try to do everything in order. So, first, before dealing with "for all $1\le k\le n$", let us deal with your previous comment: "It seems the irony that the generated list is numbered with Roman numerals is lost on ChatGPT", which is factually incorrect. It would be nice to finally see your response to that. | |
| Apr 21, 2023 at 13:51 | comment | added | Iosif Pinelis | @MartinArgerami : Thank you for your comment and the upvote. | |
| Apr 21, 2023 at 12:18 | comment | added | Martin Argerami | I upvoted this answer. But regarding arithmetic, addition and subtraction are easier with Roman numerals than with the usual decimal notation (of course, for numbers small enough for the Roman numerals to be used). | |
| Apr 21, 2023 at 5:58 | comment | added | Carl-Fredrik Nyberg Brodda | @IosifPinelis I would read it aloud, and let its grammar be, as "for all $k$ between $1$ and $n$", or a small variation of this -- because that's how it's used. I'm a descriptivist, not a prescriptivist. | |
| Apr 20, 2023 at 23:49 | comment | added | Iosif Pinelis | @Carl-FredrikNybergBrodda : Also, can you let me know how you propose to change grammar so that the phrase "for all $1\le k\le n$" could be read aloud, and how would you read it aloud? Please try to be quite specific here. | |
| Apr 20, 2023 at 23:38 | comment | added | Iosif Pinelis | @Carl-FredrikNybergBrodda : I can respond to your latter comment. Before I do so, I would like to see your feedback to/acknowledgment of my response to your previous comment: "It seems the irony that the generated list is numbered with Roman numerals is lost on ChatGPT." Do you agree with that response of mine or not? | |
| Apr 20, 2023 at 22:12 | comment | added | Carl-Fredrik Nyberg Brodda | @IosifPinelis Honestly I don’t see why “for all $1 \leq k \leq n$” is ungrammatical. Yes, it’s ungrammatical if you read the individual symbols one by one, after another. But in languages we can combine symbols to change their grammar (for a relevant and a bit silly example, the grammar of “AI”, being a noun, is quite different from the grammars of the article “A” and the pronoun “I”). So why not let the grammar of “$1 \leq k \leq n$” in this context be as we all use it? I also don’t see how the ambiguity of $n$ is somehow more of an issue in $1\leq k \leq n$ than it is in $k \in [1,n]$. | |
| Apr 20, 2023 at 15:02 | comment | added | Iosif Pinelis | @TimothyChow : Indeed, this notation may be confusing. | |
| Apr 20, 2023 at 4:26 | comment | added | Timothy Chow | Your remarks about random variables reminds me that I find it very confusing that conditional entropy $\mathrm{H}(Y|X)$ is an expected value and not a random variable. | |
| Apr 20, 2023 at 2:31 | comment | added | Iosif Pinelis | @NateEldredge : Yes, there may be some very subtle difference here. Yet, both phrases "for all $1\le k\le n$" and "for all $k\in[1,n]$" are bad, each in its own way: the former one is ungrammatical (and possibly ambiguous: is $n$ fixed here or not?), while the second one misses the condition that $k$ be an integer (which you apparently meant here). A good replacement would be "for all $k\in[n]$" (assuming that the notation $[n]$ is universally understood, or explained) or "for all integers $k\in[1,n]$" or "for all $k\in\{1,\dots,n\}$". | |
| Apr 20, 2023 at 1:30 | comment | added | Nate Eldredge | Replacing "for all $0 \le x \le 1$" with "for all $x \in [0,1]$" is one thing. But replacing "for all $1 \le k \le n$" with "for all $k \in [1,n]$" is another. | |
| Apr 20, 2023 at 1:14 | comment | added | Iosif Pinelis | @JochenGlueck : On yet another thought, I would suggest that the term "paraphrasing" does not seem appropriate here at all. Indeed, paraphrasing means restating a grammatical phrase into another grammatical phrase of the same or close meaning. But the phrase "for all $0\le x\le1$" is not grammatical in the first place. So, what you were trying to do was, not paraphrasing, but repairing the phrase, turning it into a grammatical one. | |
| Apr 19, 2023 at 23:48 | comment | added | Iosif Pinelis | @JochenGlueck : I don't think this is paraphrasing. This is an example of using synonyms, which indeed makes the speech more lively and less repetitive. Similarly, we use "belongs to" or "is a member of" in place of "is in". | |
| Apr 19, 2023 at 22:50 | comment | added | Jochen Glueck | @IosifPinelis: While paraphrasing might be avoidable in most cases, I don't think that it should be avoided. E.g., in a course on ordered Banach spaces I recently wrote "$\forall x ,y \in E \; \exists z \in E: \; z \ge x,y$" on the blackboard. While writing it I said "For all $x$ and $y$ in $E$ there exists $z$ in $E$ which is an upper bound of $x$ and $y$". A few seconds earlier I had used the expression "which dominates both $x$ and $y$" for the same property. I find this more lively and more intuitive than sticking to "such that $z$ is greater or equal than $x$ and $y$" throughout. | |
| Apr 19, 2023 at 18:35 | history | made wiki | Post Made Community Wiki by Asaf Karagila♦ | ||
| Apr 19, 2023 at 17:41 | comment | added | Iosif Pinelis | @JochenGlueck : If you do want to emphasize the compactness, I think it is much better to just write: "for all $x$ in the compact interval $[0,1]$", instead of "for all $0\le x\le1$" or even instead of "for all $x\in[0,1]$". However, I think "paraphrasing" is hardly ever unavoidable or warranted; mere expanding of symbols according to their definitions should almost always suffice; actually, at this point I cannot think of any exceptions. | |
| Apr 19, 2023 at 17:32 | comment | added | Iosif Pinelis | @LSpice : Thank you for your further comments. I still think we should avoid contradicting grammar, especially when it is so easy to do. | |
| Apr 19, 2023 at 17:29 | comment | added | Iosif Pinelis | @JPMcCarthy : I think ChatGPT can be a useful tool, and it has given me correct answers about some rather nontrivial (to me) math on a couple of occasions, even though on most other math-involved occasions it was mostly nonsense. The question is how to make it useful. In this case, for this posted question, it did supply some useful ideas, even if some of them were misleading. So, I did try to sort them out. | |
| Apr 19, 2023 at 17:28 | comment | added | Jochen Glueck | For this reason, I'm quite open to writing things which I might also need to be paraphrased a bit when reading them aloud. The point made in @LSpice's comment is also a good one, I think. | |
| Apr 19, 2023 at 17:27 | comment | added | Jochen Glueck | @IosifPinelis: I do agree that your suggestion (paraphrase every symbol) works well in this example. I also think, though, that it's a fine line between expanding the symbols according to their definition and paraphrasing. If, for instance, a compactness argument plays a crucial role, I would most likely say "for all x in the compact interval from 0 to 1". If "the closed interval from 0 to 1" is the definition of [0,1], I thus already started pharaphasing. And I'll do so much more ruthlessly when it comes to more complicated formulas. | |
| Apr 19, 2023 at 17:22 | comment | added | LSpice | Re, I think that @JochenGlueck's suggestion was, in my awkward paraphrase, that the expansions of some symbols are context dependent. For example, some people write "Let $x\in\mathbb R$" to mean "Let $x$ be an element of $\mathbb R$", but "Suppose that $x\in\mathbb R$" to mean "Suppose that $x$ is an element of $\mathbb R$"—I don't like this, but it's common. So one could argue that, in the context "for all $0\le x \le1$", the expansion is "for all $x$ such that $0\le x$ and $x \le1$". | |
| Apr 19, 2023 at 17:22 | comment | added | Iosif Pinelis | @LSpice : Thank you for comments and edits. I think we should avoid contradicting grammar, though. | |
| Apr 19, 2023 at 17:22 | comment | added | JP McCarthy | I tried to ask Chat GPT some group theory questions yesterday. Absolute garbage, wrong answers. | |
| Apr 19, 2023 at 17:14 | comment | added | Iosif Pinelis | @JochenGlueck : I don't think that we need to "anyway paraphrase what [we] read". Rather, every symbol in a math formula needs to be expanded -- according to its definition. For instance, the compact formula "$x\in[0,1]$" contains three symbols: (i) $x$; (ii) $\in$ (meaning "is in"); and (iii) $[0,1]$ (meaning "the closed interval from $0$ to $1$"). So, just expanding the formula "$x\in[0,1]$" by substituting the meanings of the symbols, we read: "$x$ is in the closed interval from $0$ to $1$", with no problems and without any need to paraphrase anything. Do you not agree? | |
| Apr 19, 2023 at 17:13 | history | edited | LSpice | CC BY-SA 4.0 |
Oops, wasn't aggressive enough about removing space
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| Apr 19, 2023 at 17:06 | comment | added | LSpice | In the construction $a x b$ for multiplication, I suppose one is meant to take implicitly $x = 1$. 😄 \\ I think one issue is that we surely all can read "for all $x \le 1$" or "for all $x \ge 0$" with no hiccoughs, and logic, in contradiction to grammar, suggests that surely these two together are fine. \\ MathJax is not as smart/aggressive as TeX about swallowing whitespace, so command definitions must end on the same line as the following text to avoid a spurious blank space. I edited accordingly. | |
| Apr 19, 2023 at 17:00 | history | edited | LSpice | CC BY-SA 4.0 |
Removed spurious space; -- -> —
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| Apr 19, 2023 at 16:58 | comment | added | Jochen Glueck | I tend to disagree with the reason why you describe "for all $0 \le x \le 1$" as bad notation. (Though one could argue that it is ungrammatical from a purely linguistic point of view.) Whenever I read mathematics aloud I will anyway paraphrase what I read to make it easier to digest. Your example illustrates this quite well: when reading "for all $x \in [0,1]$" aloud I would rather say something like "for all x in the interval from 0 to 1" rather than reading literally what is written on the paper. Similarly, I would read "for all $0 \le x \le 1$" aloud as "for all x between 0 and 1". | |
| Apr 19, 2023 at 16:33 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Apr 19, 2023 at 15:54 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Apr 19, 2023 at 15:52 | comment | added | Iosif Pinelis | @Carl-FredrikNybergBrodda : Actually, the lower-case Roman numerals (in parentheses) are mine. I find them convenient for enumeration and separation of items of comparatively small length. But, of course, it is hard to do arithmetical or other numerical operations with Roman numerals. | |
| Apr 19, 2023 at 15:42 | comment | added | Carl-Fredrik Nyberg Brodda | It seems the irony that the generated list is numbered with Roman numerals is lost on ChatGPT. | |
| Apr 19, 2023 at 15:36 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Apr 19, 2023 at 15:31 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Apr 19, 2023 at 15:21 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Apr 19, 2023 at 15:10 | comment | added | Ben McKay | I have seen x used for cross product on MO today. | |
| Apr 19, 2023 at 15:05 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Apr 19, 2023 at 15:02 | comment | added | Iosif Pinelis | @DenisSerre : Interesting. | |
| Apr 19, 2023 at 15:01 | comment | added | Iosif Pinelis | @LSpice : You are quite right. | |
| Apr 19, 2023 at 14:49 | comment | added | Denis Serre | Do you know why Romans did not invent neither algebra, nor statistics ? Because they considered $X$ as a constant (equal to $10$). | |
| Apr 19, 2023 at 14:48 | comment | added | LSpice | What is the precedence of $\mathsf E$? It looks like you're saying $\mathsf EXY$ should mean $\mathsf E(XY)$, but probably $\mathsf EX+Y$ shouldn't mean $\mathsf E(X+Y)$. | |
| Apr 19, 2023 at 14:42 | history | answered | Iosif Pinelis | CC BY-SA 4.0 |