aa r X i v : . [ m a t h . HO ] S e p WHO OWNS THE THEOREM?
MELVYN B. NATHANSON
Abstract.
Epistemological and sociological questions about ownership of math-ematical and scientific discoveries.
Simple questions: If you prove the theorem, do you own it? Can you forbidothers to use or even cite it? Can you choose not publish the theorem? Can yoube forbidden to publish it?What is theorem? A mathematical statement may be true. It is true whetheror not there is a proof. Without a proof, we do not know if it is true. A theoremis a true mathematical statement that has a proof.Suppose there is a true mathematical statement, and you prove it. Now it is atheorem. It is “your theorem.” In what sense might you own it? Can or should atheorem be considered the private property of its discoverer, who may or may notchoose to publish? If you own the theorem, can you license it or rent it? Can youinsist that anyone who wants to use or apply the theorem must pay you to do so?If you publish the theorem in a refereed journal, or post it on arXiv, or explain itin a seminar, or submit it to a journal, then everyone knows you proved it. Whendoes “your theorem” become part of the public library of proven mathematicaltruths that other researchers can freely use to prove new theorems?If you need a result to prove a theorem, and know that the result is true butthe discoverer has not announced or released it publicly, is it ethical (of course,properly citing the discoverer) to use that “unpublished” result in the proof? Is itethical for you not to prove the theorem because it requires a result that is truebut is being withheld by its “owner”?Suppose you find out that someone has proved a theorem, but has not revealedit to the world. Maybe you have even seen the proof, and checked it, so you aresure that it is correct. Even though it has not been published, you know that it isa mathematical truth. Can you use it in a paper, even though the discoverer mightnot want the result to be known? Does the prover of the theorem own it enoughto prevent other mathematicians from using it?The notion of “owning a mathematical truth” is, in part, connected with ca-reerism in academic life. What might be called “vulgar careerism” is endemic andnot necessarily vulgar. Many mathematicians hide what they are working on soothers will not “scoop” them, will not use “their” ideas to prove a theorem beforethey do. Perhaps it is not sufficient to give proper attribution. Maybe the author isan untenured assistant professor who wants to deduce more results from the theo-rem, publish more papers, and get promoted. Maybe the author thinks it will leadto a proof of the Riemann hypothesis and earn the million dollar prize from the
Date : September 18, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Philosophy of mathematics, sociology of mathematics.
Clay Mathematics Institute. Some mathematicians admit that they discuss theirideas about how to solve the Riemann hypothesis only after they are convincedthat the ideas will not work.It used to be that, every year, permanent professors in mathematics at the In-stitute for Advanced Study in Princeton would appoint a visiting member to betheir “assistant.” Long ago at the Institute, there was a permanent member whorequired his assistants to promise not to reveal to anyone what he was workingon. When I learned this, I was shocked. It was antithetical to everything I be-lieved about science. I was also naive. I had not understood that for many peoplemathematics is a competition. In 1977, the National Security Agency decided that publication of cryptographicresearch would endanger national security, and wanted to require that professorswho wrote papers in cryptography would have to send them for pre-publicationreview by the NSA and not submit them to journals without NSA approval. Atfirst, NSA hoped for voluntary compliance, but also considered making this a legalrequirement. This did not happen, and, after considerable contentiousness anddebate in the mathematical community, prepublication review by the NSA fadedaway and seems not to be an issue today [1, 2].Secrecy in mathematics is less important than in other sciences. Mathematicalresults rarely have commercial value. Like many mathematicians, I don’t care ifmy theorems are “useful.” I only hope that I have not made mistakes, that theproofs are correct, that the “theorems” are theorems and are interesting. I uploadpreprints to arXiv as soon as they are written, before I send them to a journal. I amhappy if someone uses my results. But this does not answer the central question.Mathematician A proves a theorem, and mathematician B learns about it. MaybeB reviewed A’s NSF proposal. Maybe A submitted the manuscript to a journal andB refereed it. Can B use the theorem (as always, with proper attribution) in a paperbefore A has published it?For me, the answer is clear. Here is an analogy. Legally, you cannot sequence aplant or animal DNA strand and patent the sequence because you did not createthe sequence. God created it, and you only discovered it. Similarly, mathematicaltruths exist, and mathematicians only discover them. If you discover a theorem,you have the power, the privilege, and, perhaps, the right not to reveal it to anyone,but if, somehow, someone learns of your result, knows that a certain mathematicalstatement is true, then that person has the right to tell the world and to apply itto obtain new results, with or without your consent.Can you own a scientific truth? Can you hide a scientific truth? These areethical questions, and, in the covid era, not only in mathematics.
References [1] S. Landau,
Primes, codes, and the National Security Agency , Notices Amer. Math. Soc. (1983), 7–10.[2] W. Diffie and S. Landau, Privacy on the Line, Updated and Exanded Edition: The Politics ofWiretapping and Encryption , MIT Press, Cambridge, MA, 2007. I was once Andr´e Weil’s assistant at the Institute. He did not impose a secrecy oath. I still do not understand why, for some mathematicians, getting medals in high school andcollege competitions is a core part of their self-esteem.
HO OWNS THE THEOREM? 3
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