I am currently reviewing a theoretical CS paper that is incredibly technical and mathematical. I’ve been reviewing the paper for longer than a year now, and I am finally recommending the paper to be accepted as I believe that it is a significant push in a certain problem in my subfield.

However, there are numerous instances in the paper where the researcher evaluates closed form formulas for integrals/summations that are incredibly difficult to do by hand (they involve extremely complex substitutions/transformations/algebraic manipulation/ and quite frankly, non “human-like” steps). Just as an experiment, I plugged in one of the integrals into maple and it gave me the exact same steps in their proof. I found three such occurences where the integrals/summations proofs are simply copied from maple.

I’m inclined to call out the researcher on this, but I have no evidence except what I’ve presented here. What makes me more frustrated is that I wouldn’t have changed my acceptance decision in the slightest had he simply quoted the integral/summation with a note that it was simply evaluated/verified in Maple. Should I just skip this from my review and focus on more important aspects of the paper?

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    Is their a reason why you can't simply write "It appears like a CAS was used to perform steps x, y, and z. The authors should cite the software."
    – Roland
    Aug 10 at 14:11
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    Why do you need such evidence? If they decide to make that argument, so be it. They might decide to simply add the citation.
    – Roland
    Aug 10 at 14:14
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    Why should a commercial product need a citation? Any commercial product? Word? Win11?
    – Buffy
    Aug 10 at 14:34
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    There was some discussion below (on one of the answers) that we should perhaps separate the two issues of (1) whether the authors did anything wrong, and (2) whether "calling them out" is an appropriate response to a perceived transgression. At this point I think the question stands as-is, just documenting this discussion for posterity.
    – cag51
    Aug 12 at 23:54

Maybe it’s a generational thing, and maybe my mind is playing tricks on me, but I get the impression that young people today are very quick to “call out” other people for actions that they perceive as moral transgressions of various sorts, where in fact those actions range a wide spectrum from actually ethically problematic behavior, to mild sloppiness, ignorance or negligence, to “offenses” that exist strictly in the imagination of the person doing the calling out.

This can get a bit tiresome.

My suggestion is that you try to adopt the habit of assuming the best about people’s intentions until they leave you no choice. If they do X, don’t immediately jump to the worst possible explanation as to what their motivation was for for doing it. And don’t immediately assume that everyone has (or should have) the same beliefs or moral code as you about every single thing, including such head-scratching, esoteric questions such as whether Maple should be cited for the steps of evaluating an integral. (Edit: this paragraph is addressing some things said by OP in comments which were later deleted, so it may sound off the mark to some. I’m leaving it here to preserve some of the context of my original answer.)

Is it appropriate to point out that it might be advisable to cite Maple for those calculations, if indeed the authors of the paper used Maple? Yes, sure, that certainly sounds like a way of making yourself helpful as a reviewer. But “calling out”? No, I think that phrase and action are sadly overused these days, and should be reserved for much more problematic circumstances.

Edit: I gave some thought to the question of whether it’s actually necessary to cite Maple for the calculations you mentioned. The thing to keep in mind is that there is a well-developed theory of symbolic integration, with standard algorithms that are implemented in the major CAS packages. I don’t know enough about Maple to know whether it also has some additional, especially novel, proprietary algorithms that make it possible for it to evaluate integrals that other packages can’t. But in the current case that doesn’t sound so relevant; the main point is that these days there is no great honor to be had among mathematicians for the “discovery” of how to evaluate a given integral, assuming the steps do not involve some highly non-standard transformation or technique. These days if someone says “I evaluated this integral, here are the steps”, no one will care very much how they found the steps, since it is assumed that they relied on known techniques (and software tools implementing those techniques) unless that’s clearly not the case. The situation is very much analogous to writing out the numerical value of the square root of some number without explaining what brand of calculator you used. No one thinks that calculators deserve special mention for knowing how to calculate a square root.

The bottom line is, if the Maple integral evaluations used standard techniques that are widely known in symbolic integration circles, I don’t think the researcher has a special duty to cite the software. It may be nice of them to do it, and it may even make their paper a bit better if they did, but from an ethical point of view, I don’t see the lack of a citation as a major issue.

  • Comments are not for extended discussion; this conversation has been moved to chat.
    – Massimo Ortolano
    Aug 11 at 16:52
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    The continued discussion here raised a potentially good point about the need to separate (1) reviewing CASs and (2) the social aspect of "calling out" colleagues; I have consolidated this discussion as a new comment on the main post.
    – cag51
    Aug 13 at 0:00

If the proofs are short enough one can verify them by hand, then I view using a computer algebra package similarly to using a calculator or dictionary. I think citing is optional.

On the other hand, if someone says "routine use of the trig formulas and standard calculus" will solve this equation, and it turns about to be 200 pages of computer generated formulas are needed, then the source code should be available and the computer algebra package cited.

You are free to suggest the paper would be better if it mentioned the use of a computer package. I do not think "calling them out" is appropriate.

Also, if there are more natural proofs, then the authors should be encouraged to spend a little time looking for them. There are lots of proofs that are generated by humans that are awkward, unnatural and hard to follow. It is not always worth it to spend a month to find a nicer proof.

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    Agree especially with your last paragraph. A colleague reported to have submitted an (later influential) paper, only to get reviewer's response to simplify the (very technical and convoluted) proofs. He responded with "Yes, happy to do so. Do you see any way how?" The reviewer gave up and accepted the proofs as is ... Aug 10 at 17:09
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    The proofs of each of these integrals is almost 2 pages of really heavy workout. My main disagreement is that the authors claim that these proofs are their contributions, when in fact, they are not. They are quite literally copy pasted from the CAS with minor "explaining notes" added in-between steps.
    – Coconut
    Aug 11 at 4:18
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    Nobody knew the proof until they used the CAS. They discovered it. If the proof is human-checkable, who cares? You can suggest a better writeup, but many will find it wrong if you say they are being dishonest. Aug 11 at 4:29
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    Alright, I reflected on this a bit and I think you're right. I might've overreacted. While the exact steps are not entirely theirs, the result was not known prior to them proving the lemma.
    – Coconut
    Aug 11 at 4:43
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    Ok, but on the other hand --- it seems you find this paper hard to read, and you think a mention of the CAS could help. Say something like that, and see if you can get this to be a better paper. Aug 11 at 4:50

As @Roland says, there's no harm in stating your observation. You can easily frame your suggestion so that there's no problem if you turn out to be wrong, e.g.

The paper presents complex algebraic solutions without comment (steps x, y, and z) that were likely done with the aid of a computer algebra system (such as Maple). If this is indeed the case, the authors should cite the software used.

As @DaveLRenfro comments, you could also point out that you were unable to reproduce/verify the results by hand/without a CAS (but maybe the authors are just way better at integration than you are ...). If the authors want to double down and say "no, we did these by hand", that's their problem.

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    If the OP is worried about the authors' doubling down, maybe say instead: "The paper presents complex algebraic solutions without comment (steps x, y, and z) that the reviewer was not able to verify except with the aid of a computer algebra system such as Maple. If the authors used a computer algebra system for these, then the authors should cite the software used." Aug 10 at 14:32
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    You probably also want to include a note to the editor why you think this is the case
    – trikPu
    Aug 10 at 14:33
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    @DaveLRenfro This should be an answer. Note that maybe other Computer Algebra systems may give the same answer, so it does not have to be Maple. OP should refrain from specifying a particular product. Maybe they used Wolfram Alpha, which would mean it's even an internet service. Aug 10 at 17:06
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    @BenBolker You did indeed, I overlooked it. Given that, maybe put it in brackets. I might also consider adding reproducibility as an additional argument why this is important. Aug 11 at 0:06
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    @Captain Emacs: What I wrote is nearly identical to what Ben Bolker wrote, so I don't think what I wrote merits a separate answer. My slight revision was only to indicate how the reviewer might introduce a bit of added pressure on the author(s) to be more explicit about how the computations were done. Aug 11 at 10:10

I would say it really depends on the circumstances.

I am one of the developers of Maxima, and I also routinely use Maxima in calculations employed in physics papers. Maple, too, and occasionally, Mathematica.

Computer algebra can be used for trivial things. By way of an admittedly trivial example, I may run into a trigonometric expression like 4*sin(x)^3-3 sin(x), which looks vaguely familiar, so without giving it a second thought, I'd feed it to Maxima, radcan(trigreduce(4*sin(x)^3-3*sin(x))), and presto, I get -sin(3*x) which goes into my paper. Would I mention that I used Maxima for this? No more than I'd mention that I used a calculator to calculate, say, 10/7 = 1.4286. Same goes for relatively straightforward integrals and other simplifications, even if the intermediate results are difficult for humans to keep track of.

The one potential issue with this is that computer algebra systems are programmed by humans (like yours truly) and may not be free of bugs. But that is kind of a user beware thing; it is best to check critical results by recalculating them in more ways than one, perhaps using more than one computer algebra system. In the end, it's no different from making a mistake when doing calculations on paper: always double check your critical results!

In short, in these situations I would not cite the use of computer algebra or expect, as a reviewer, an author to cite its use.

The situation is very different when the use of the computer algebra system is more elaborate. When it's not a single command line but a lengthy, elaborate program in that computer algebra system, designed to set up, solve, and simplify a problem. In that case, yes, I'd very much expect not so much a cited reference to Maple (or whichever CAS is used) but perhaps the actual computer algebra code included in the paper as an appendix with a brief explanation or, if it is too lengthy to be included, then deposited in a public archive and referenced. Otherwise there is no reproducibility: how do we even know that the program (perhaps written by a researcher with limited programming experience as his expertise lies elsewhere) is even correct?

From the wording of the question, however, I get the impression that it is a case of the first variety: the question itself suggests that once the reviewer recognized that the solution was likely obtained using Maple, he was able to obtain the same result with ease.

So no, the simple fact that a result was copied from Maple is not a reason to cite Maple, just as copying a result from the display of a calculator is not a reason to mention that calculator.


It seems like you are treating this as a homework assignment where the instructor said "all work must be done by hand", but then someone went and did all of the work with a calculator. While doing the work with a calculator would be wrong in that instance, here I don't see what the issue is.

Coming from a physics/mathematical modeling background, I have seen countless papers write out derivations of equations where huge steps are left out or only briefly described. Never once have I ever seen a paper say "we did all of this work by hand", or, "we used Maple, Mathematica, etc. to make sure we did our math correctly". You just assume the math was done correctly, and you don't really care about how it was explicitly done.

The only reason I can see this mattering is if the method of "evaluating closed form formulas for integrals/summations" was the whole point of the paper, but the results were actually obtained a different way. e.g. the authors claim to use their new method/algorithm, but really they just plugged things into Maple, then that would definitely be dishonest. But if the equations are just being used for other things and the derivation is not the point of the paper, then I honestly don't see how they got the derivation done matters at all.

So I will agree with others; there is no need to call anyone out. From what I can tell nothing bad is going on. The most I would do is just suggest giving credit to any software that was used.

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    My experience in physics is that more reasonable phrasings like "checked using Mathematica up to N=20" or "evaluated using Mathematica" are used occasionally. That said, my hope/impression is that the refereeing standards for proofs, perhaps especially in pure math and theoretical CS, are a little stricter than "just assuming the math was done correctly"...
    – Anyon
    Aug 11 at 17:46
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    In my view, how the authors obtained the derivation does matter for auditing purposes. Suppose that the authors of a given paper do write something along the lines of "we used function f in Maple". Suppose further that, years later, it is found that Maple's function f improperly implemented some symbolic algorithm. If the TeX files were in arXiv and their authors had the good taste and the foresight of documenting which functions they used, then it should be relatively easy to discover all papers that used function f and notify their authors that they might want to revise them. Aug 11 at 17:50

Common mathematic techniques are not special because of who carries them out. If the steps are valid, that's all that matters. It looks like those are all mechanical transformations that by themselves don't make the paper special. It's just usual mathematical drudgery that has to be done correctly, but otherwise is of no importance to the novelty of approach. That Maple or some other CAS actually "did it" doesn't change anything whatsoever. The steps in a proof stand on their own.

Now, the CAS may be implementing some special algorithm for a tricky class of integrals etc., that may be unique, new enough and citeable, but 1) the help pages for the functions used would probably document that if indeed the algorithm was special enough to warrant that, and 2) figuring out that any particular algorithm actually got used requires way more investigation than is warranted in such a paper - CAS systems won't normally tell you what algorithms they used (here: "algorithms" = "citations to where said algorithms got published first"). You'd have to ask a specialist in CAS algorithms to take a look at what was done by the CAS, and whether there's anything there that's cool/interesting enough that a citation would be warranted. Otherwise you may as well be citing dozens of foundational CAS algorithms. So that's not really feasible.

Citing the CAS as a whole is like "citing" someone who checked your computations. You can thank or acknowledge them - sure, but you won't be citing it.

I think that the idea of calling someone out for using a CAS the way others use a calculator is mostly preposterous because it is absolutely unproductive. At the end of the day, nobody gets any value out of it. The reason for citations is to let others learn more about what the paper is built on, and to give due credit. When it comes to using common technical software, credit is usually given in hard currency... and that's all that's needed unless the software is specialized. CAS isn't really. Entirely mechanical, "garden variety" tools like CAS are not exactly a dime a dozen, but how would citing them help, and moreover how would "calling someone out" over it help anyone? It just sours things up. I fully expect that anyone wanting to reproduce or fully grok the results in the paper will have to know enough that knowing how to use a CAS, and which CAS may be sufficient for the task, will be the least of anyone's concerns.

If the paper built some derivation specifically on the "extraneous" aspects of the CAS system, e.g. if its approach integrated some aspects of, say, Mathematica's language and idioms - that'd be different. But here, that wasn't the case. It doesn't look like the paper's discoveries used scripts/programs written in Mathematica/Maple, leveraging the key aspects of the environment. They just did the integrals and such using Maple. Maple itself doesn't matter here it all.


One of the frustrations of academia is that as a reviewer, you often invest large mental and creative effort and time as an anonymous referee to verify and even make understandable important results, for no credit or acknowledgement. The metaphors "panning for gold" and even "pearls before swine" apply.

It's therefore very easy to get frustrated and grumpy at the nth instance of the authors making it unnecessarily hard for you and future readers.

From the discussions in comments on the question and other answers, it sounds like once the frustration has dissipated, a comment in your review along the lines of the following might suffice:

Lemma 3.21 [or whatever] is very technical and hard to follow. I was able to verify the proof using a CAS (Maple), which precisely generated equations [...] through [...], leading me to believe this proof was generated with CAS in the first place. It would be easier for the reader to follow -- and provide a more realistic impression of the specific contribution brought by this paper -- to note a CAS was used, and perhaps cite the specific CAS/version.


Each person's use of that famous Canadian1 mathematics software named after a tree2 requires agreeing to the company's EULA. While not all provisions of all EULAs are enforceable in all jurisdictions, the text of the EULA does communicate intent and wishes of [REDACTED TREE]Soft3.


The License of all Software hereunder is subject to the express restrictions set forth below in addition to the restrictions imposed by the applicable License Option, Installation Type and Order Confirmation. Without the express written permission of [REDACTED TREE]Soft, YOU shall not, and shall not permit any Third Party to:


(h) use [REDACTED TREE]Soft’s name, trade names, logos, or other trademarks of [REDACTED TREE]Soft, or any of its Affiliates or Licensors, whether in written, electronic, or other form, without [REDACTED TREE]Soft’s prior written consent;

So under the rules of [REDACTED TREE]Soft's EULA, nobody using their software should ever mention the name of the software (that famous Canadian mathematics software named after a tree, which has a trademarked name) or the company name without prior written consent of [REDACTED TREE]Soft.

While I am not one to determine if that contract term is enforceable where you live, it does make it really clear that [REDACTED TREE]Soft doesn't want you to use their name. Doing so in a publication would be against their clear request in their EULA, unless you first cleared the use with [REDACTED TREE]Soft itself.

Note that the contents of your workbook:

(a) reproduce, transmit, modify, adapt, translate or create any derivative work of, any part of the Software, in whole or in part, except for any content developed by users in Maple Worksheets/Documents that are not part of an electronic book Software product or as otherwise expressly permitted in this Agreement;

are explicitly permitted to be shared without the consent of [REDACTED TREE]Soft.

At a moral level, [REDACTED TREE]Soft clearly does not want its name mentioned. So users of that famous Canadian mathematics software named after a tree should respect its wishes and not mention its name.

Unless, of course, you have prior written permission.

It is just being polite.


2 Not an oak.

3 All uses of [REDACTED TREE]Soft in the quote below are redacted by me, not in the original document. The tree is also not a beech.

  • Contractual agreements with third parties don't release authors from their obligations to readers. If academic standards require that the step in question be explained, the authors would need to say something like "we used a computer algebra system which, for legal reasons, we cannot name". If the editors don't think this is enough information for readers, then the authors have to either get permission from the vendor, or redo their work in a way that they are allowed to explain in sufficient detail. Aug 14 at 17:52

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