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I have read about many great researchers in various fields who were not involved in academia throughout their lives. Some were involved in academia only part of their life, but almost invariably, their greatest work came after a period of solitude and isolation from others. In fact, most great mathematicians, for example, were rather isolated figures. Bernhard Riemann and Isaac Newton come to mind, but others like James Clerk Maxwell were also isolated during their periods of research. More recently, Andrew Wiles isolated himself for around 7 years and didn't speak about his work to anyone while he was busy solving Fermat's Last Theorem. (I have mentioned only mathematicians because those are the ones I have read about the most.)

Hence, I am failing to see how being in academia/a university position aids one as a researcher, practically speaking. It seems as though the greatest research must come from within the individual due to an intense and personal love and desire for the subject, rather than due to collaborating with others or being given a better position or whatever. In fact, such things seem to be the opposite: detrimental. They would make the person focus not on the subject but on paltry things like gaining more and more money or a better reputation.

My question is simply: How does being in university position (e.g. professor) aid the researchers who are at the highest level in their field? I do not mean being involved in academia, because one can be involved (eg attending seminars and lectures) without having a position (please correct me if I am wrong, as I myself am not actually in academia).

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    Andrew Wiles was a faculty member at Princeton while he worked on his proof. Getting paid to do research is generally considered a good thing by those wanting a roof over their heads and food on the table. – Jon Custer May 14 '18 at 16:19
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    Well for starters it gives money to live. Being a Mathematician is a full time job for most people... – Evariste May 14 '18 at 16:20
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    For starters: research how Riemann, Newton, Maxwell, etc. supported themselves. – GEdgar May 14 '18 at 17:25
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    Researches can't live in a hole and at the same time build a large hadron collider all by themselves. – henning -- reinstate Monica May 14 '18 at 17:42
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    And note Yitang Zhang didn't work at Subway when he proved his major result --- he was at the math department at the University of New Hampshire. – Aaron May 15 '18 at 16:26
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Beyond the obvious benefits of financial stability, one must also consider the possibility that not all researchers are solitary creatures. This is especially true in the non-mathematics parts of STEM fields, which frequently require collaboration to achieve meaningful results.

Moreover, there is the additional issue of the amount of work that one person can get done relative to the number of ideas and projects that they want to explore. For instance, right now, there are about a half-dozen different project ideas that I'd like to work on, and I could probably come up with a dozen more if I had the time and space to do so. But I only have a finite number of hours that I can work, which means that I have a few options:

  • find people who are willing to work on those ideas by getting a grant and hiring graduate students or postdocs,
  • do it myself, or
  • leave the work undone.

The last option is unappealing, and the second is basically impossible without burning myself out after a few months. So that leaves the first option, which is difficult to do if you're not affiliated with some sort of research institution.

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    Mathematics also requires collaboration often. It's more common for people to work alone on a topic, but it's very rare that a researcher will write all their papers alone. – user9646 May 15 '18 at 10:06
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What follows does not answer so much the general question posed, so much as address the examples offered in the question. All of the question's examples answer the question.

Professionalstability and remuneration are conducive to engaging in the sort of committed, intense, and prolonged thinking and studying that results in deep conceptual advances. Very few professional contexts outside academia offer this sort of environment.

Andrew Wiles was already a tenured full professor at Princeton already when he did his work on the Taniyama-Weil conjecture and Fermat's last theorem. He studied at Oxford and Cambridge, wrote a PhD with the famous number theorist John Coates, and, prior to his work on FLT, had coauthored papers with Coates and the famous number theorists Karl Rubin and Barry Mazur.

Newton, Riemann, and Maxwell were university professors and all of their work was conducted in what were at the time the standard institutional contexts for such work, with the provisos that all three were recognized as exceptional talents, Maxwell and Newton ascended professionally more rapidly than was the norm, and all Riemann and Maxwell encountered the sorts of ordinary professional mishaps in changing jobs (and wars, and illness - Maxwell had smallpox and Riemann died from tuberculosis) and the like that all of us experience in some measure.

None of these people was particularly isolated in a professional or social sense. Maxwell and Newton occupied prestigious professorships at young ages. Newton served in parliament, was warden of the mint, and was president of the Royal Society! Riemann probably had the most difficult road professionally, but he was helped by Gauss, who recognized quite fully Riemann's talent, and he interacted with the other prominent mathematicians of his era and locale (Jacobi, Dirichlet, etc.).

These are all extreme examples of people of unusual talent and who worked unusually hard. The academic context allows them to support themselves economically while taking the risks necessary to pursue difficult lines of research that might fail. For every one of them there many others of similar talent and work ethic who pursued ideas that developed less fruitfully. The academic context allows this mass of talented, hard-working people to develop ideas that might fail, and without this the ideas that succeed might not be developed at all.

Riemann and Wiles stand out for having published relatively few papers, all of high quality. This is almost impossible without the stability provided by excellent work conditions of the sort that can only be found in an academic environment (Wiles published nothing that appeared in print from 1991-1994 - this is impossible for someone who does not already have a permanent academic job!). The sort of positive isolation conducive to deep thinking is almost impossible in non-academic environments.

In general, being in an academic environment is important in the current time for the following practical reasons: availability of doctoral students, access to research funding, access to financing for travel, and access to other researchers. Additionally, in institutions that function well (a minority), the work conditions facilitate creativity and intellectual risk taking.

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In the modern era (say, the last 50-100 years) non-professional contributions to science and mathematics have been extremely rare. (When they happen, you read about them). Doing science is a full time job, and being employed at a university or (the right kind of) industrial research lab provides you payment for spending much of your time doing research. Most science is also collaborative --- and universities provide collaborators, in the form of PhD students, postdocs, and other faculty. The modern examples you give --- Andrew Wiles, and Yitang Zhang --- were both employed at universities (although Zhang is a closer match for what you want, since he was employed as a lecturer at the time rather than a tenure track professor). A recent example of amateur mathematics making an impact is the solution to the Hadwinger Nelson problem: https://www.quantamagazine.org/decades-old-graph-problem-yields-to-amateur-mathematician-20180417/

So it certainly occasionally still happens that someone who is not a professional mathematician employed in an academic setting solves an open problem. But when it happens it is news, and I know of nobody who has done this more than once.

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It seems as though the greatest research must come from within the individual due to an intense and personal love and desire for the subject, rather than due to collaborating with others or being given a better position or whatever.

This assumption is both not necessarily true, and not necessarily true for both all people and all fields. My best work, for example, has emerged from collaboration with others, passing discussions, etc.

Indeed, I'm not entirely sure the image of the lone genius laboring away in isolation, fueled only by passion, hasn't done more harm than good.

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Many greats answers on academic career of famous and incredibly talented people.

At least German universities are what they are (a mixture of research and teaching) because they root on some philosophical principles that say they should mix research and teaching. While not willing to dive deep, the idea is that research facilitates better teaching (on to the frontiers!) and that teaching facilitates better research (why was that like this again?).

So, putting research institutes and company-based R&D aside, if you are a tenured researcher at a (German) university, you do some teaching.

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