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What recent discoveries have amateur mathematicians made?

Can anyone give historical examples of people outside of mathematics that have contributed to the literature? I'm particularly interested in academics from distant fields (humanities, social sciences) who have developed their hobby into a noteworthy contribution, but would also be interested in those from nearer fields (sciences), where the math was not part of their area of research, and those not affiliated with academia at all.

I'm a math graduate student with hopes to contribute to philosophy or humanities someday. I'm just curious how impenetrable different disciplines are, and realised I know of no examples like this.

Please forgive me if this is a duplicate, I couldn't find another like it, and think others may find it interesting as well.

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marked as duplicate by quid, Daniel Moskovich, Andrés E. Caicedo, Loop Space, Willie Wong Sep 4 '11 at 12:20

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

This question is quite similar… Out of curiosity, given your motivation I am surprised you ask about 'outsiders' in math, rather than the reverse, what you seem to hope doing. – user9072 Sep 4 '11 at 1:49
Freeman Dyson didn't bother with getting any PhD AFAIK. Of course this is a rather special case... – Steve Huntsman Sep 4 '11 at 2:14
Insofar as theoretical computer science is a part of mathematics, Chomsky should qualify. Despite being a linguist, his work helped shape modern theoretical computer science. – Logan Maingi Sep 4 '11 at 2:28
As perhaps best exemplified by Russell (but also a number of less well known others), logic and certain schools of analytic philosophy are close enough that it is not always possible to tell them apart. Of course, there are other schools of analytic philosophy who see Russell's approach to philosophy and in particular the resemblance to mathematics as fundamentally wrong-headed. – Alexander Woo Sep 4 '11 at 2:48
Incidentally, without a PhD $\neq$ amateur. In Italy, for instance, the PhD programme was instituted in 1980 only (with the exception of one special school). Therefore most Italian mathematicians at the end of their career do not possess one, even though they hold tenured positions. – Federico Poloni Sep 4 '11 at 19:49

Buffon(Georges-Louis Leclerc, Comte de Buffon; 1707 – 1788) is a towering figure in biology.
As a mathematical hobbyist he invented geometric probability theory.
His method of calculating $\pi$ by throwing pins on a floor ruled by parallel lines and recording the number of throws in which the pin crosses one of those lines must surely count as one ot the most extraordinary ideas ever in mathematics.

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To answer your second paragraph rather than the first, the best way to break into a field other than your own is to form a collaboration with someone already in the field. Doing it on your own could be too difficult as you will be unknown and your writing will feel alien to people who grew up in the field (even if it is technically excellent). Once you establish a reputation as a collaborator, you will be better placed to strike out on your own if you wish.

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Raymond Quenau, the French writer, published

Raymond Queneau, « Sur les suites s-additives », Journal of Combinatory Theory, 12, 1972, pp.31-71.

with an introduction by Gian Carlo Rota. This maybe does not qualify as "notable" but still is not so bad for an amateur...

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To answer the question in the title: Andy Gleason never bothered with a PhD. One could speculate his service to the US Navy in WWII was the reason, and that ordinarily he would have gone through the "ordinary" channels. He became famous for his contribution, along with Deane Montgomery and Leo Zippin, to the solution to Hilbert's fifth problem.

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This doesn't really count. Gleason was a Junior Fellow at the time where this was an alternative to the Ph.D. - – François G. Dorais Sep 4 '11 at 3:16
@Francois: Thanks for this, I wasn't aware what the Society of Fellows was. There is no formal equivalent in the English system. – WetSavannaAnimal aka Rod Vance Sep 4 '11 at 6:47

Edward Witten is a winner of the Fields Medal and he's a physicist. Nor so far from mathemathics but he doesn't have a Math PhD.

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Maybe this doesn't count :) See Peter Woit's evidence (in "Not Even Wrong") for why he thinks Witten is a superintelligent alien, not human :). Seriously, though, this is a good example, also supporting Brendan Mackay's answer, as I understand Witten worked closely with mathematicians once the relevance of his work to Donaldson theory became clear. – WetSavannaAnimal aka Rod Vance Sep 4 '11 at 6:52
Quite off-topic, but Rod Vance's comment reminded me of: – Willie Wong Sep 4 '11 at 12:22

Hilary Putnam is a philosopher who played a key role in the solution of Hilbert's 10th problem.

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This falls under my Russell comment up above. At the time Putnam was working in an area of analytic philosophy quite difficult to distinguish from logic. He has since concluded that his former approach to philosophy was quite wrong-headed. – Alexander Woo Sep 4 '11 at 17:25

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