What did Grothendieck mean by "the capacity to be alone" in the context of mathematical research? I'm confused about a statement Alexander Grothendieck made in his book Récoltes et semailles.

In fact, most of these comrades whom I gauged to be more brilliant
than me have gone on to become distinguished mathematicians. Still
from the perspective or thirty or thirty five years, I can state that
their imprint upon the mathematics of our time has not been very
profound. They’ve all done things, often beautiful things, in a
context that was already set out before them, which they had no
inclination to disturb. Without being aware of it, they’ve remained
prisoners of those invisible and despotic circles which delimit the
universe of a certain milieu in a given era. To have broken these
bounds they would have to rediscover in themselves that capability
which was their birthright, as it was mine: The capacity to be alone.

Why did Grothendieck say “the capacity to be alone,” and how does this phrase apply to researchers?
As this is a translation, the words &quot;capacity to be alone&quot; may not be the best way to phrase it. Perhaps this is a more literal translation, but from context I might write it as &quot;independence&quot;.
He is saying that other people who he perceived as more capable individuals (&quot;more brilliant than&quot; he) as researchers were successful, and yet were constrained by doing work similar to others. Perhaps they worked on incremental solutions to problems within a general framework in the research community.
He implies that the status quo (&quot;invisible and despotic circles which delimit the universe of a certain milieu&quot;) is a constraint on creative work.
He is suggesting that his own impact has been more profound or more revolutionary because he has been able to be more independent from others, to make his own path either in the problems he approached or in the solutions he found, and that he thinks the other people he talks about would have had similarly profound impacts if they had been less focused on the work or opinions of others.
Who knows whether he is correct that this is how researchers should think, it's just one opinion. I think it's somewhat dangerous to assume that because an approach worked for someone that you should take their advice that it would work for everyone. There is a lot of survivorship bias in that those who go their own path but are not successful are seldom heard from - who would read their book?
I'm not a mathematician and am not familiar with Grothendieck's work. Just from his Wikipedia page it seems he became a bit reclusive and didn't engage with the broader community later in his career. From Wikipedia which talks about the work that you quote from:

In the 1,000-page autobiographical manuscript Récoltes et semailles (1986) Grothendieck describes his approach to mathematics and his experiences in the mathematical community, a community that initially accepted him in an open and welcoming manner but which he progressively perceived to be governed by competition and status

It could be that he misjudges the ways in which his earlier work that became so influential benefitted from that of others around him, and misjudges the profundity or impactfulness of the work that he later pursued in solitude. Just because someone has a genius in mathematics or any other field does not necessarily mean they are good at assessing the reasons behind their own success; self-reflection is an entirely separate skill set.
For what it's worth in contrast, another famous mathematician wrote, &quot;If I have seen further it is by standing on the shoulders of Giants&quot;, which seems to be a bit of a different philosophy (though I may be treating Grothendieck uncharitably, as it seems his quarrel is less with the work of others and more with the academic establishment, but it still speaks to the value or lack thereof of independence/solitude in academic work). Google Scholar uses the motto &quot;Stand on the shoulders of giants&quot; in recognition and tribute to this sentiment.