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Amongst philosophers, Socrates is an example of a genius with a great influence on human history who never wrote anything. Almost all facts which are known about his revolutionary ideas are written by his students like Plato and Xenophon.

Question: What are examples of genius mathematicians in modern or ancient eras whose ideas had a great influence on development of mathematics but never wrote or published anything?

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    $\begingroup$ Genius Mathematicians who Never Published Anything - Yeah, I know, I'm slow with the typing... Just have patience. $\endgroup$ – Lucian Dec 12 '14 at 2:00
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    $\begingroup$ @Lucian I am waiting eagerly... :-) $\endgroup$ – user180918 Dec 12 '14 at 2:01
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    $\begingroup$ Do classmates count? You know, those guys who won't do problem sets but "know their math really well". $\endgroup$ – Mark Fantini Dec 12 '14 at 2:16
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    $\begingroup$ This question would fit very well on History of Science and Mathematics $\endgroup$ – Danu Dec 12 '14 at 7:49
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    $\begingroup$ I generally dislike the idea of "mathematical geniuses". Great ideas from great people come from hard work in all cases... $\endgroup$ – Dirk Dec 12 '14 at 18:48
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There are influential posthumous papers by authors who did not publish them.

Galois had an immense influence on algebra because of the publication of something he wrote the night before he died in a duel. He gave us the word "group".

Thomas Bayes had an influential posthumous paper in which he found the conditional probability distribution of a random variable $P$ whose marginal distribution is uniform on $[0,1]$, given the observation of the number of successes in $n$ trials that are conditionally independent given $P$, where the conditional probability of success on each trial, given $P$, is $P$. He did publish things while he lived.

Mary Cartwright never published her proof of the irrationality of $\pi$. A question she set on an examination was to fill in its details. We know of it only because Sir Harold Jeffries included it in an appendix in one edition (and not in other editions) of one of his books, and he leaves the impression that he knew of it only because of its occurrence on that examination.

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    $\begingroup$ For those who don't know, Galois was less than 21 years old when he died in that duel! $\endgroup$ – David H Dec 12 '14 at 18:55
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    $\begingroup$ @Michael Hardy: could you indicate the gist of Dame Cartwright's proof? (Although back in graduate school I heard her lecture about her work on dynamical systems growing out of the British radio problem, I never before heard about the proof you mention.) $\endgroup$ – murray Dec 12 '14 at 22:14
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    $\begingroup$ Here is is: en.wikipedia.org/wiki/… ${}\qquad{}$ $\endgroup$ – Michael Hardy Dec 12 '14 at 23:46
  • $\begingroup$ @murray : I failed to "notify" you when I posted the item above. Now you're notified. $\endgroup$ – Michael Hardy Dec 13 '14 at 1:12
  • $\begingroup$ No, Galois developed his ideas on groups years before and had written manuscipts. What he wrote the night before the duel was letters to urge other people to publish his work postumously. $\endgroup$ – Somos May 9 '17 at 19:28
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The words "never" and "anything" are somewhat restrictive (and we know that absolute statements are always wrong). Nowadays, you can't be recognized at all when you're not publishing. Additionally, the term "Genius" is hard to define...

One person came to my mind is Grigori Yakovlevich Perelman. He hardly published anything, did not even defend his dissertation, and was one of the few leading researchers at his institute who only was a "PhD candidate". Of course, strictly speaking, he did publish something, particularly his proof of a generalization of the Poincaré conjecture. But this was not published in some mathematical Journal, but only on arXiv.

I think that someone who casually finds a proof for a generalization of a conjecture that dozens of mathematicians had been working on for 98 years, and doesn't give a ... care about the formal process of scientific publications qualifies as a genius, and is close enough to "never publishing anything" to be mentioned here, at least...

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    $\begingroup$ (+1) Thanks for your nice answer. Just a logical paradox: "absolute statements are always wrong!" is an absolute statement itself! $\endgroup$ – user180918 Dec 12 '14 at 11:07
  • $\begingroup$ @AliSadeghDaghighi I like paradoxes (maybe I should have made clear somehow that this was intentional) $\endgroup$ – Marco13 Dec 12 '14 at 11:11
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    $\begingroup$ @Marco13 We got you the first time. Now its so overt. That always spoils the charm. $\endgroup$ – zxq9 Dec 12 '14 at 14:26
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    $\begingroup$ Perelman published quite a few papers in a usual way. Even before the whole story about Poincare's conjecture he was offered a number of positions in different US universities. $\endgroup$ – Artem Dec 12 '14 at 18:44
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    $\begingroup$ "Hardly published anything" is not quite right. I don't have access to MathSciNet now but on Google Scholar I found more than 10 published by Perelman (also with well known coauthors). I guess his total paper count is about 20. $\endgroup$ – Dirk Dec 12 '14 at 18:45
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Riemann published but 15 papers in his lifetime. One of my professors once told me, perhaps apocryphally, that one of those papers contained an error, and that his distress over the error led him to poor health and eventually his death.

After his death, his housekeeper "cleaned up" his papers, losing who knows how many of his mathematical ideas forever.

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    $\begingroup$ Are the downvotes because of the anecdote, or because 15 is too large to be in the domain of "nothing". $\endgroup$ – Emily Dec 12 '14 at 19:51
  • $\begingroup$ I'd guess for the latter. $\endgroup$ – quid♦ Dec 17 '14 at 13:42
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Pierre de Fermat-Some of his works and theorems were published by his son. His works, including Fermat's last theorem $(a^n+b^n\neq c^n \,\forall n>3,a,b,c,n\in\mathbb{Z})$ were not published until his death.

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  • $\begingroup$ Was going to correct you and say "some." And people such as Pascal knew of his work - much of Fermat's work was communicated in private letters to other scientists. $\endgroup$ – Gyu Eun Lee Dec 12 '14 at 2:17
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    $\begingroup$ Did Fermat have much to say about his last theorem? Other than the proof being too big to fit in the margin of his paper? ;) $\endgroup$ – eigenchris Dec 12 '14 at 19:05
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    $\begingroup$ In those days there was nothing resembling "publication" in the sense of the last 250+ years. $\endgroup$ – paul garrett Dec 12 '14 at 19:37
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I think Thales of Miletus is the most obvious answer, although perhaps a bit boring. From wiki:

Thales involved himself in many activities, taking the role of an innovator. Some say that he left no writings, others say that he wrote On the Solstice and On the Equinox. (No writing attributed to him has survived.)

He is said to have basically invented mathematics, or rather the axiomatic proof-based approach to mathematics.

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Joshua King could be an example of a genius mathematician who never gave a lecture and published almost no papers but one.

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    $\begingroup$ The linked page seems to me to suggest that he was not a genius mathematician; he certainly doesn't seem to have done any mathematics after he graduated Cambridge, which I would argue disqualifies him as a mathematician no matter what his title is. $\endgroup$ – cpast Dec 12 '14 at 8:18
  • $\begingroup$ @cpast I think at that time, it was clear to people around him that he is a genius mathematician and a true substitute of Newton and this was the main reason for appointing him on such a high academic position. $\endgroup$ – user180918 Dec 12 '14 at 10:48
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    $\begingroup$ Again, not how I'm reading it. The way I read it, he showed promise in examinations, but while people at the time thought he was a genius mathematician, the reputation was entirely undeserved, because for all practical purposes he was not a mathematician, and he did no mathematics after being first in his class at university. $\endgroup$ – cpast Dec 12 '14 at 15:08
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To echo others, the question leaves a lot open to interpretation; however, my vote would have to go to Gosset, the inventor of the "t Distribution."

William Sealy Gosset (13 June 1876 – 16 October 1937) was an English statistician. He published under the pen name Student, and developed the Student's t-distribution. (wiki)

Given that Gosset published under a pen name, it could be argued that he never "published" .

Many people are familiar with the name "Student" but not with the name Gosset. In fact Gosset wrote under the name "Student" which explains why his name may be less well known than his important results in statistics. He invented the t-test to handle small samples for quality control in brewing. Gosset discovered the form of the t distribution by a combination of mathematical and empirical work with random numbers, an early application of the Monte-Carlo method. (http://www-history.mcs.st-andrews.ac.uk/Biographies/Gosset.html)

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