Earlier Thoughts that Inspired Grothendieck (2014) [pdf]

A cursory search yields that actually mental health is correlated with academic achievement (and thus mental health problems must be anticorrelated):

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4443903/

Of course, this is "academic achievement" as in getting a 4.0 at Stanford, not creating a new field of mathematics...

I imagine your question would be hard to investigate as many of these high achieving mathematicians are rare outliers and would defy statistics on population averages...

i have a personal, not necessarily scientific opinion on this. i think that certain minds are so above normal life that they are able to creatively push the bounds of our knowledge. now, since life is a flawed beast of chaos and disorder, these minds that have high creative but yet logical horsepower are not able to deal with the shortcomings of humanity. they could also fall into this trap of "everyone says i am wrong, but they were wrong before when i was right, so it's the case again". i think it's these facts coupled with the fact that depression can be described as people who understand the world as it truly is but can't handle it or ignore it like everyone else.

have you ever dealt with someone so illogical it just drives you mad? now think if everyone you met felt like that.

i actually don't know much about godel, but his work on logic, computation, and general relativity seems to be of the flavor "the world may not be as you think it is". that type of work could take its toll, and something did.

What makes you think Grothendieck had mental health issues?

General relativity, or maybe some stuff that Ramanujan figured out? But probably other people would have eventually figured those things out if they thought about them long enough. In general I think the answer is no. There's a fascinating book called The Sociology of Philosophies that explores this question. The author's main idea is that intellectual ideas arise from competition and variation among different intellectuals who are related to each other through intellectual networks and that only ideas that are creative, but not too creative, get remembered and extended, refined, or rebutted and passed on to later generations of scholars.

Einstein came up with the idea that light would behave like particle. He published the paper in 1905 where he eloquently explained his thinking and gave three examples demonstrating it. One of the examples was the photoelectric effect. He won a Nobel price for that discovery.

>Energy, during the propagation of a ray of light, is not continuously distributed over steadily increasing spaces, but it consists of a finite number of energy quanta localised at points in space, moving without dividing and capable of being absorbed or generated only as entities.

Plank had explained black body radiation as it would originate from quantified oscillators 1900, but nobody except Einstein believed that light could propagate as energy quanta. Only after Compton effect was discovered in 1923, everybody saw that Einstein had been on the right track all along.

Being the only physicist who saw it for almost 20 years, even when everybody has read your explanation and takes you seriously is a sign of highly original mind.