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I have a hard time understanding statements such as:

Scientist X discovered Y in 1960 and was subsequently rediscovered by Scientist Z in 1980.

How does one prove that plagiarism has not taken place? This is also extensively observed in some really old math theorems, chemistry and so on. How does one prove that the similar work produced was as a result of one’s own independent work and not resulting from another’s. Even if something was re-discovered subsequently, why is it even given merit?

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    One doesn't prove that plagiarism has not taken place. Accusers prove that it has taken place. – Scott Seidman Dec 8 '17 at 19:37
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    If late-rediscovery were any sort of academic crime, we'd pretty much have to knock every researcher who reports anything. The idea that most work's novel is one of those early naivities that a life in research will dispel. – Nat Dec 9 '17 at 1:18
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    I’ll just leave this here: Examples of Stigler’s Law of Eponymy – JeffE Dec 9 '17 at 13:48
  • @Nat I'd put that a bit differently: Rediscoveries don't happen very often, because not very many people discover stuff for the first time that is fourty years ahead of their time. ;-) – Karl Dec 9 '17 at 16:37
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    It's perfectly legit to stand on the shoulders of giants and make a nice selfie. If you only later find out that mountain was in fact a forgotten giant, nobody is going to think worse of it. – Karl Dec 9 '17 at 16:43
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Remember that research was a lot harder to track down before the Internet Age. We couldn't as easily do literature searches because databases were much more restricted in scope and scale. Therefore, we can guess that keywords might not have found the earlier results.

In chemistry, it's possible that alternate methods could be used to synthesize a molecule, which would certainly justify its republication. In mathematics, an alternate, more elegant route for the proof or justification would also merit "rediscovery."

It's only plagiarism if the same methods and techniques were used after consulting the old material: in other words, only if it's a straight-up reproduction.

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    Plagiarism isn't merely the same methods and techniques because independent rediscovery is possible, in which case the relevant question is, who was first, not did one copy the other. (This often arises in patent disputes, for example.) An essential element of plagiarism is that it's not independent and involves presenting someone else's ideas or work as your own. – Nicole Hamilton Dec 8 '17 at 19:06
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    In many cases, such discoveries were also arrived at from different directions, and the connection was not seen until later. – Jon Custer Dec 8 '17 at 19:30
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    For what it's worth, I've encountered many rediscoveries in mathematics. A typical example is described in the first two paragraphs of IV. MOST C-INFINITY FUNCTIONS ARE NOWHERE ANALYTIC in this 9 May 2002 sci.math post, where I discuss the apparently independent rediscoveries of the same result made by Morgenstern (1954), Christensen (1972), Darst (1973), and Ramsamujh (1991). – Dave L Renfro Dec 8 '17 at 19:59
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    @NicoleHamilton In prior industrial projects for, say, chemical plants, I've found the same inventions repatented over and over again over the past 150 years. But it's always in a slightly different context, using slightly different terminology, in a slightly different commercial venture, such that it seems to go unnoticed. – Nat Dec 9 '17 at 1:30
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    It's even worse for algorithms, mathematics, logic, and any other highly accessible field that a lot of people do. Like xkcd quipped (in the mouse-over text): "Some engineer out there has solved P=NP and it's locked up in an electric eggbeater calibration routine. For every 0x5f375a86 we learn about, there are thousands we never see.". Publishing something involves a ton of work and translation into a field's understanding; but the basic problems themselves have been solved a million times before. – Nat Dec 9 '17 at 1:33
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In mathematics genuine rediscovery seems to me quite common, and it is usually plain that the rediscovery is not plagiarism for reasons related to its content and exposition.

Often the rediscovery occurs in a substantially different subfield than the original discovery and is motivated by different considerations than those that motivated the original discovery. In other cases the result is essentially the same, but the techniques used to prove it are different. In other cases the results can be proved to be equivalent, but this is not self-evident from their statements, or to their authors (or others), and it requires some work, or at least some insight, for people to realize that something is being rediscovered. Sometimes a result that is now considered important in and of itself was proved in passing as a supporting lemma in an old paper, and only once the rediscovery has taken place is it recognized that the result had already appeared. Many results are forgotten - at the time they are published there does not exist an audience prepared to understand their significance - later this audience comes to be and the result is rediscovered - still later someone in that audience realizes that the result had been obtained much earlier. Finally there is what might be the most common pattern - at the time the relevance of the discovery was not recognized or the work of the person who made it was not paid much attention (for various reasons, e.g the worker published little, wrote in a strange language, died early, etc.). Others come along, and with similar motivations pursue the same lines of research. It is usually clear that such rediscovery is not plagiarism simply from the style and organization of the exposition. More often the results are not exactly the same in the so-called rediscovery, although there may be some overlap.

Each of the scenarios described can be illustrated by numerous examples, but as doing so takes some work (to get the details straight) I'll limit myself to the following two that are more or less fresh in my mind.

Here's an example. It is a theorem proved by C. Lebrun in 1987, and generally attributed to Lebrun, that there exists no integrable complex structure on the six-dimensional sphere that is orthogonal with respect to the standard Riemannian metric. The existence (or not), on the six-dimensional sphere, of an integrable complex structure (with no further condition) is a famous open problem in differential geometry (and one for which many false "proofs", both affirmative and negative, have been published, even quite recently). The theorem proved by Lebrun is an important partial result in that it shows that the most natural ways of proving an affirmative resolution fail. Recently the open problem has been revisited and it has been realized that it (actually something more general) was proved in 1953 by A. Blanchard (surely some specialists knew this earlier, but the broader "community" seems not to have (I learned it from a comment made by R. Bryant on MathOverflow); in a different context, a different result from the same paper of Blanchard was rediscovered by A. Sommese, as he explains in the his MathSciNet review MR0397030 of a paper of S.T. Yau). Blanchard's article has probably been overlooked partly because it was written in French and Blanchard only published a few papers, so probably his work was not well known to later workers, particularly as the use of French in mathematics declined; more likely, people weren't ready to appreciate its import it at the time, when the general understanding of integrable complex structures was still undeveloped. (A review of this particular theorem can be found in this article of A. Ferreira.)

Here's another example (really of independent discovery). There is a mathematical structure now known under the names left-symmetric algebra, Vinberg algebra, pre-Lie algebra, and chronological algebra (among others - the multitude of names for one object is already indicative of the sort of lack of communication that often underlies rediscovery and independent discovery). It was discovered essentially simultaneously in three radically different contexts in the early 1960s. E.B. Vinberg introduced such algebras under the name "left-symmetric" in his study of homogeneous convex cones (these are finite-dimensional left-symmetric algebras). M. Gerstenhaber introduced the graded (so slightly more general) version of these structures, under the name "pre-Lie" algebras, in his study of deformations of algebras. They are also closely related to the B-series introduced by J. Butcher in the context of the analysis of Runge-Kutta type methods for the numerical solution of ordinary differential equations (here it is infinite-dimensional pre-Lie algebras that are relevant). The "chronological algebra" terminology arose in the work of A. A. Agrachev and R. V. Gamkrelidze on control theory. These contexts are superficially quite different and the particular algebras that appear in the different contexts have different characters. There was for a long time essentially no overlap between people working in these areas. It took many years for people to assimilate that essentially the same mathematical structure was being studied in parallel in different contexts, with different points of view and different techniques. In retrospect there are good reasons for the appearance of this algebraic structure in these apparently diverse contexts, and understanding why the same structure was appearing in these apparently disparate contexts is itself quite illuminating.

I view these kinds of rediscovery and independent discovery as useful. They often shed new light, illuminate complementary points of view, or bolster confidence in complicated arguments. It is desirable to give credit where credit is due, but the obsession with novelty for its own sake in publication sometimes ignores that methods and exposition can be novel even if the results are not, and that this can be useful for better understanding and utilizing the results.

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    An example I discussed in my dissertation (and it's not the only such example I discussed there) involves the symmetric derivative of a function from the reals to the reals, where difference quotients f(x+h) - f(x-h) / 2h are used. A standard elementary real analysis exercise is to prove that if the ordinary derivative exists finitely at a point, then the symmetric derivative also exists there (and has the same value). The absolute value function (at the point x=0) shows the converse fails. Since there exist nowhere differentiable continuous functions, a natural question is (continues) – Dave L Renfro Dec 10 '17 at 11:56
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    whether there exist nowhere symmetrically differentiable continuous functions. The credit for the first such function is pretty much universally given to Filipczak (1969), and Kostyrko (1972) gave a Baire category argument that “most” continuous functions are nowhere symmetrically differentiable. (In fact, I know of no published or even unpublished references that say otherwise besides my dissertation and a handful of places where I’ve mentioned this online.) (continues) – Dave L Renfro Dec 10 '17 at 11:57
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    However, in 1958 Petrův (1958) gave both an explicit example of such a function and a Baire category proof that “most” continuous functions are examples. Moreover, Petrův’s actual results were even stronger than these two later published results, because he used symmetric difference quotients in which 2h is replaced by an arbitrary scale function, so the result becomes one about having at no point a preassigned and uniformly applied pointwise modulus of symmetric continuity. – Dave L Renfro Dec 10 '17 at 11:58

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