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The modern standard is that "between two or more independent discoverers, the first to make formal publication is the legitimate winner", colorfully described as publish or perish. This was not always the case. In 16-th century it was common for scientists to hide their discoveries and assert their priority by challenging others to solve problems they could solve, Tartaglia's story being a famous example. Nor was it the case in the 17-th century, when Newton and others coded their discoveries into anagrams instead of publicizing them. It is interesting that in some early versions of the patent law "actual use of the invention was deemed adequate disclosure to the public" too.

Wikipedia says that the publish or perish rule came into effect "as soon as modern scientific methods were established" and mentions Newton and Leibniz, which is ironic since in the calculus priority dispute there never was any doubt that Leibniz published first. The point of contention rather was whether he saw Newton's private letters sent to Collins and Oldenburg prior to that. Even in 19-th century Gauss is often credited with discovering non-Euclidean geometry based on his private papers, even though he not only never published his findings but made a point of not even mentioning them publicly for fear of "the scream of beotians".

When did publish or perish become the "ironclad" priority rule? Unlike patents and copyrights it was never codified into law, so what was the mechanism of its adoption? Did Academies and Royal Societies play a role? Was it influenced by developments in patent and copyright law? When did universities start basing their hiring decisions on publications as they do now?

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The main reason for "publish or perish" is not securing priority. (Newton did not publish much, but was very much concerned about priority. Some such people exist even now).

The reason is that "research" is required nowadays from professors at the universities. Their salary and even employment depends on the "research" they produce, and the simplest way by which the bureaucrats can evaluate our research is the number of publications and the rating of the journals.

Another indication of research is the ability to obtain grants. Grants bring to professors and researchers a lot of benefits, and in many cases obtaining them is the condition of employment.

The current system developed gradually, mainly in 20-th century and it continues to develop. It is related to science becoming a mass enterprise.

In the middle of 20-th century in many places it was still normal to defend a PhD and to obtain a permanent teaching position. After that your only duty was teaching.

In the first half of 20-th century there were famous mathematicians who published less than 10 papers in their lifetime. For example, Eduard Helly, whose name is known to every math student.

Nowadays one is required to publish several papers every year and in some places they even say that 3 papers per year is not enough. Otherwise, even if you have a permanent position, you will not receive grants and will be poorly treated by the administration.

Grants system was introduced on a massive scale in the second half of 20-th century. The expression "publish or perish" appears at the same time.

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    $\begingroup$ Here is some strong circumstancial evidence for your claim in the last paragraph. Perhaps edit it in? $\endgroup$ – Danu Dec 5 '14 at 8:09
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    $\begingroup$ Perhaps, for priority purposes publish or perish is not so ironclad even today. Perelman never technically published his proof of geometrization, and the papers he posted on arxiv had many (minor) gaps that nonetheless took several years to fill in. However, when Yau tried to be formal about it it wasn't well-received en.wikipedia.org/wiki/Manifold_Destiny#Controversy. $\endgroup$ – Conifold Dec 12 '14 at 8:56
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    $\begingroup$ Please don't take Perelman as an example. He also resigned from his job. His behavior is an exception, not a rule. Most of us have to make a living of our research. $\endgroup$ – Alexandre Eremenko Dec 12 '14 at 14:03
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    $\begingroup$ What is Eduard Helly known for, on the undergrad level? $\endgroup$ – Nikolaj-K Dec 25 '14 at 1:53
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    $\begingroup$ Look "Helly's theorem" (convex geometry) and "Helly selection theorem" on Wikipedia. Both things belong to the undergraduate curriculum, but I do not claim that every undergraduate in every university is taught this:-( $\endgroup$ – Alexandre Eremenko Dec 25 '14 at 7:29

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