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If I read a math proof in a paper that requires payment to read, is it legal to rewrite the proof with different words, phrasing, and order, then post the proof publicly in a website (with a citation)? Or we are forbidden from publicly sharing math proofs until the paper's copyright expires?

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  • This post doesn't answer the question but is semi-relevant math.meta.stackexchange.com/questions/12232/copyright-and-proof
    – Stuart F
    Commented May 21, 2022 at 16:29
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    This is similar to the question of if a rewrite of a computer program would violate the copyright of the original; looking into that might give you some ideas.
    – usul
    Commented May 22, 2022 at 1:42
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    Probably the question is whether your work would be considered a derivative work. If so, it would infringe the original's copyright, but might still be legal under fair use. In any case, I think most mathematicians would be very comfortable doing this without legal concern and I think it happens all the time on blogs, perhaps arxiv, etc.
    – usul
    Commented May 22, 2022 at 1:51
  • For what purpose is this? If the only purpose is to be able to publicly post a copy of the proof and/or to evade the copyright protections, you are a lot less likely to fall under Fair Use (US) allowances. Commented May 22, 2022 at 12:31
  • From the standpoint of academic practice and ethics (which is not necessarily the same as what is legal!), I think that deliberately rephrasing, so that you change only the words but the actual steps remain identical, would be frowned upon. What's more acceptable is to read the proof, understand the high-level ideas of it, then set it aside and write your own proof from scratch based on the understanding you have gathered. Include a citation of the original. Textbook authors do this all the time. Commented May 22, 2022 at 19:31

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A proof can be protected by copyright. The underlying facts of math cannot. But if one has copied details of the order of the proof, or of the selection of theorems to use, and if several other choices would have been possible, then the new proof may constitute a trivially modified copy, or a derivative work, and in either case making of it might be copyright infringement.

However, making and distributing a copy, even with no changes at all, for purposes of comment and criticism, might be fair use in the US, fair dealing in the UK or some other parts of the Commonwealth, or fall under an exception to copyright in other countries (these generally vary significantly by country). This is usually a very fact-driven question.

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    "A proof can be protected by copyright." I think more care and nuance is needed here. When a mathematician says "a proof" they usually refer to the general ideas of the steps, not any particular expression in a specific form. 8 pages of English prose and one chalkboard full of Greek symbols could be considered the same proof.
    – usul
    Commented May 22, 2022 at 1:38
  • the selection of theorems to use, then rewriting proof is impossible without infringing copyright because rewriting proof require the same selection of theorem (rewriting is not the same as creating new proof), does this means i cant rewrite proof without copyright infringement ?
    – LLL
    Commented May 22, 2022 at 9:23
  • @LLL: That goes to the merger doctrine, and possibly implicates cases like Baker v. Selden (or your jurisdiction's equivalent, if any). It's a bit of a gray area, and I would suggest retaining counsel if you actually plan to do this.
    – Kevin
    Commented May 22, 2022 at 18:48
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Copyright law only protects "original expression". The originality of a mathematical proof lies in the underlying idea, but that idea is not itself protected by copyright. What is protected is the expression of the idea. Typically, however, a mathematical proof contains no original expression, it contains a standard symbolization of that idea. The surrounding text in ordinary (typically minimal) could be protected, except for formulaic utterances like "Therefore:", "It follows that"... As the US Copyright office says, mathematical principles are not protected by copyright. Facts are not protected, and every mathematicians that I have known considers their proofs to be facts.

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    A major proof typically consists of a more-or-less long chair of facts (often formulae) combined with reasons why each fact follows from a prior fact, or an axiom of the system., The selection of which facts to use in a proof, which prior facts to derive them from, and which rules of reasoning to use may be protectable. Note that major theorems often have many proofs, For Example see the detailed discussion of the many proofs of Euler's Theorem (V+E-F=2 for a polyhedron) in Proofs and Refutations* by I. Lakotos. Commented May 21, 2022 at 19:25

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