# Is the following copyright infringement or scholarly analysis? Both? Neither?

Here is the question:

Showing ${\rm Aut}(Q_{2^n})\cong{\rm Hol}(\Bbb Z_{2^{n-1}})$ for $n>3$

I have asked many questions like it.

The copyright section of the book states:

This work may not be translated or copied in whole or in part without the written permission of the publisher [. . .], except for brief excerpts in connection with reviews or scholarly analysis.

My view is that it is scholarly analysis; however, @markvs disagrees. See the comments of the linked question.

• Perhaps Law.SE……? Jan 8, 2022 at 2:51
• A little searching led me to law.stackexchange.com/questions/56278/… , law.stackexchange.com/questions/70656/… Jan 8, 2022 at 2:57
• Neither of those answer my question, @CalvinKhor, since the textbook states that scholarly analysis is permitted. Jan 8, 2022 at 3:03
• I think you should contact SE directly to get an official answer. I don't think we can rely on answers and comments on Math Meta for such legal issues. Jan 8, 2022 at 3:04
• An author cannot create an "anti-license" to bar legitimate uses that do not rely on permission. Fair use, fair dealing, and other exceptions to copyright law still apply the same as they always do.
– Nij
Jan 8, 2022 at 3:05
• @Shaun I agree; sorry for the terse comments! Jan 8, 2022 at 3:15
• Mods and employees of SE often reply to meta posts, @user21820HATESSMOKING-HATS. Jan 8, 2022 at 3:50
• I know that, but I think it's still better to go directly or at least post on the main Meta SE rather than on the site Meta. Jan 8, 2022 at 4:17
• Take a look at stackmod.blog/2012/04/april-newsletter-2 Jan 8, 2022 at 19:16
• springer.com/gp/rights-permissions/obtaining-permissions/882 may be a good place to look into.
– Pedro Mod
Jan 9, 2022 at 16:00
• "I have asked many questions like it." Are you implying a systematic program to ask exercises from a particular copyrighted work? The amount of reproduction involved (relative to the size of the work as a whole) is one of four factors when copyright suits are heard, But there is no "safe harbor" percentage that one is allowed when no permission is granted by the copyright holder. Jan 9, 2022 at 17:28
• Just a few exercises from the book I got stuck on, plus similar questions from other books in the past, @hardmath. Jan 9, 2022 at 17:41
• When I search on Questions here about solutions to exercises in Robinson's book, I find more than a dozen were posted in the last year, and all of those Questions were asked by you. So Readers might reasonably worry about your purpose, which is also one of the factors considered in a fair use defense. Of course the math is not copyrightable, but the fixed form of exercises in a published text is copyrightable. Jan 9, 2022 at 17:51

It is the opinion of the Math SE moderation team that this question does not represent a clear violation of copyright law as we understand it. If you believe that our judgement is in error, please use the "Contact" link at the bottom of the page to raise the issue with the Community Moderation team and/or SE staff.

– Xander Henderson Mod
Jan 22, 2022 at 13:18

Beware "Internet lawyering" in all of its forms. Having said that - and to be clear, I am not your attorney or giving you legal advice - I work in this area and can comment generally on what I have seen in the US.

In the US, a plaintiff asserting copyright infringement on the basis of a discussion of a single exercise in a math textbook (even if that discussion involves verbatim recitation of the statement of a problem) would be highly likely to lose, and very early in litigation. This isn't to say that a vexatious plaintiff could not cause a defendant to incur legal expenses in responding to the lawsuit. I just don't see how they would win the lawsuit. (Please note, in America, the default rule is that both sides pay their own attorneys' fees regardless of who wins. This is not the default in other jurisdictions.)

There are two main reasons for this. First, empirically speaking, factual matter - that is, content close to some kind of objective reality, where the form of how something is described might be partially dictated by bounds that are not subject to the whims of the writer - is not afforded broad protection in the US. Second, there is the doctrine of fair use.

None of this has anything to do with words that might be put by the publisher in the front of the book. I don't know of any case in which a court has found these types of recitations meaningful in assessing the boundaries of liability. It is certainly possible for somebody to become forbidden (e.g. by contract) to do something that they otherwise would have been able to do. In the US, the question of whether printed matter is effective to create legal obligations is a matter of state law, not federal law, and so can have 50 potentially different answers. But I have never seen boilerplate in the front of a textbook litigated. My guess is that I haven't seen it litigated because in most states such language would be regarded as meaningless.

I should add a few cautionary words. There is a lot of commentary about what is or is not US "fair use" as though there are settled rules in this area, and there mostly aren't. (The doctrine is codified in 17 USC 107, reflecting words largely drawn from an 1800s-era opinion by Joseph Story, but these words shed very little light on how the doctrine is applied in specific circumstances). In real life, if you get sued, you can be entirely right about US fair use but may have to spend quite a bit of money to prove it. Lawrence Lessig has described fair use as "the right to hire a lawyer," and he is not wrong about that. A lot of defendants who are not in the wrong find out that it can be cheaper to pay a settlement than it is to pay lawyers who can prove that they are right. (Some countries, not the United States, have formalized rule-bound doctrines that expressly place certain activities within a "safe" realm of non-actionable behavior. Fair use in the US is not like that - every defendant has to prove their own case.)

Despite this general indeterminacy, however, academic commentary on a math exercise is the absolute core of what the doctrine exists to protect. It isn't close. If someone asked me to file a lawsuit on that basis I would be concerned about the possibility of being sanctioned for doing so. (I would cite cases, but do not do so because I do not want this comment to resemble personalized legal advice, which it is not.)

So the broader question: what happens if we are talking not just one exercise, but a lot of exercises from the same book. In the US, it is very much within the rights of the owner of a copyright in a math textbook to use copyright claims to suppress the unauthorized distribution of a solution manual to that textbook. A plaintiff would argue that even if individual exercises exist in the realm of unprotectible "fact" (which is debatable), their selection and arrangement in a course of study is protectible expression. And there is an enormous amount of authority that a plaintiff could cite in support of that point. Unauthorized solutions manuals to copyrighted textbooks are extremely likely to be regarded as infringements of copyright. (Publishers who respond to this kind of thing with legal threats, which I have seen happen, are not just bluffing or attempting to intimidate. A cautious attorney would tell a client who wanted to distribute a full solutions manual without authorization not to do that.)

In the middle, we have the question of degree: what if someone's activity on a site includes, in bits and pieces and in various places, discussion that might in some aggregate sense resemble a solutions manual. There's no one answer. It depends! Much of what renders compilations of factual or near-factual matter copyrightable in the US (namely, originality in the form of selection and arrangement) are arguably absent from a discrete series of different questions. This is true even if some search function allows a site user to review all questions by a single user, or all posts about a specific textbook. That's what I'd argue on the defense side. I don't think such a case is a slam dunk for the plaintiff, and I would suspect the motives of anybody who said that it was. Having said that, I wouldn't want to defend such a case, either.

• If I write a solutions manual where the only reference to the original book is the title, and perhaps the numbers of the questions, it is hard to see that that is copyright infringement, as I don't replicate the "content" of the book - just responding (but without "copying"). Am I misunderstanding what you said? What does "copyright" mean when you are not "copying"? Jan 10, 2022 at 14:01
• @Floris there is no general doctrinal reason in the US why the owner of the copyright in the textbook would automatically lose such a suit (and in particular, there is no legal obstacle to them filing a lawsuit and imposing costs on a defendant). The alleged "copying" in this instance would probably be framed as reproducing (albeit indirectly) enough of the substance of a sequence of exercises for it to be wrongful. This is not an automatic win for the plaintiff; there could be argument either way. But "copying" in the US legal sense does not have to involve verbatim reproduction. Jan 10, 2022 at 16:38
• thanks for that clarification. Jan 10, 2022 at 16:54
• The problem here is that it is not going to be a case User Shaun v Professor Emeritus Derek Robinson. It is going to be SE v Springer. The latter is one of the biggest scientific publishers in the World and can afford the best lawyers. So every "protected" argument will be met with a powerful counterargument. As I mentioned in my comment on main, you can sing "Yesterday" (in your shower) but if you decide to publish a portion of Beatles catalog, Sir Paul (or whoever owns it now) will sue you and will be able to afford better lawyers. Jan 11, 2022 at 9:15
• Another, although related, issue is ethical. There are certain rules developed by many generations of mathematicians, one of which is "do not steal" with very broad understanding of "steal". Basically it means: if you publish a copy of somebody's work, make sure this somebody is not against it and mention that in your publication. Mathematicians who violate these rules loose a lot of real world reputation which costs much more than the imaginary reps on SE. Jan 11, 2022 at 9:16
• I am not sure that SE would be a natural defendant in a hypothetical case like this (the reasons are complicated and unrelated to the OP's question). The most closely analogous situations I have seen involved claims brought against individuals. I do agree with the broad point that the views of an author (and a broader community) are likely to be more relevant professionally than the question of what is legally allowed to do. I also see this as orthogonal to the question. Discussing solutions to exercises is scholarship in a sense relevant to US copyright even if it violates a scholarly norm. Jan 11, 2022 at 17:06
• @markvs I cannot think of any mathematician I know that would think ill of someone studying mathematics from a Springer book and merely replicating the statements of the exercises on the internet. In fact, a lot of mathematicians I know make the draft version of their copyrighted books publicly available (from Springer, or others) so students that cannot afford the books have access to them. Perhaps your experience is different.
– Pedro Mod
Jan 21, 2022 at 16:25
• @PedroTamaroff: It is good that a moderator finally responded. I am a mathematician, and the Editor-in-Chief of a math journal. So I know something about copyright. Whether concrete mathematicians think ill or not is irrelevant here. As I said the case will be (if at all) Springer vs SE. Jan 21, 2022 at 17:39
• @markvs I am not talking too much as a moderator of the site, but as a mathematician (notice I am not employed by SE, I am merely a volunteer). What is your point regarding your credentials? I am sure most if not all people understand the issue you're raising about copyright and, at least given the responses I see here, we are all taking it seriously.
– Pedro Mod
Jan 21, 2022 at 18:47
• @PedroTamaroff: My point is that this is a copyright issue. And being an Editor-in-Chief, I have to deal with these issues all the time. I understand that you are not a SE employee, I would be interested in the SE opinion about this and I am farther than you from them. This is a problem not specific to Math SE. Jan 21, 2022 at 18:52
• @markvs Sure. I think my point is that perhaps there are better fights to fight. In particular, this is not my fight, so I'll go about my merry way. Good luck!
– Pedro Mod
Jan 21, 2022 at 18:56
• @PedroTamaroff: I do not understand why it is difficult to understand that this is not a question about individual opinions or reactions. Bye! Jan 21, 2022 at 19:04

Here are my cents of wisdom on this matter (from mathematicians' perspective):

1. SE is aware of the potential copyright issues arising from SE posts. For instance, here is what SE says in its April 2012 Newsletter, directed at moderators:

Your Role in Copyright Enforcement What do you do when someone claims an answer was copied illegally, or maybe a teacher is claiming that a student is posting test questions on your site? Here in the States, the Digital Millennium Copyright Act (DMCA) tells us very specifically how to handle claims of copyright infringement.

As moderators, you are not employees of Stack Exchange. In essence, you are free to moderate the site for content how you see fit. If you are ever notified about copyrighted material, and feel like taking it down, that’s cool… but you are not required to do so. You are not DMCA agents (we have those of our own). If you get a request for takedown, you can either remove the post OR point the person to the Copyright Policy linked below.

Here at Stack Exchange central, we have to be concerned about claims of copyright infringement that may not be valid. If WE (Stack Exchange Inc.) get notices to take down offending content, the claimants need to follow DMCA procedure and contact us with the correct form. It’s all outlined in the Terms Of Service:

Our TOS covers all the relevant info about DMCA.The claimants will have to contact us with the necessary forms and information. That makes the issue pretty simple and straightforward on your end. Delete or reply with link.

In other words, "SE will deal with the problem if a complaint is filed."

As far as individual users go, I agree with markvs: It will be "Publisher vs SE", not "... vs user."

1. There is another, ethical, issue to be taken into account (from my viewpoint):

Exercises are a valuable part of a textbook. One consideration departments/instructors make when choosing a particular textbook is presence or absence of exercises. The presence of exercises makes it easier to teach a class using a book, since one can simply assign HW problems from the textbook. Choosing HW problems from the textbook also makes teaching more uniform across different academic years.

In view of this, each time a solution to an exercise from a textbook is posted online, the value of the book goes down a bit. At some point, once too many homework problems are solved online, the department or instructor can simply decide to use a different textbook. You may not care if a publisher looses the revenue in the process, but you may want to think about the author of the textbook.

If one is a fan, for instance, of Professor Derek Robinson and his book, one should (from my viewpoint) think seriously about harming his interests by making solutions of problems from his textbook available online.

• I would say your last point holds in general, but perhaps the book in question is too specialised for this to really matter. E.g. I recall Ted Shifrin mentioning in chat he basically makes/made a very small amount of money from his great multivariable calc /intro diff geom book. Jan 11, 2022 at 11:44
• @CalvinKhor: Money is the last reason for writing a math book. Jan 11, 2022 at 13:39
• @CalvinKhor: As I wrote, the issue is ethical. Even if the amount of money is small, say, about \$100 per year, it would be unethical (from my viewpoint) to deny it to somebody in this situation, especially if this is a person whose book user likes. It might not matter to the author, but it should matter to the user of SE posting the solutions. Jan 11, 2022 at 14:25
• If you want to argue ethics then it should be stressed that an ethical mathematician would not take actions the greatly inhibit dissemination of mathematical knowledge - which such dubious DCMA claims about "copyrighted" exercises surely do. Jan 11, 2022 at 18:49
• @BillDubuque: To me, things are far from clear-cut here. I would not do this myself, but I also would not regard a mathematician (or, say, a writer, for this matter) doing this as necessarily unethical. But consider adding another answer arguing your viewpoint. Jan 11, 2022 at 20:45
• @BillDubuque Yet, publication in subscription journals (Springer, Elsevier, etc) does precisely this, and is ubiquitous! Jan 23, 2022 at 21:22