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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?

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    – cag51
    Feb 6 at 23:44
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As this is a translation, the words "capacity to be alone" may not be the best way to phrase it. Perhaps this is a more literal translation, but from context I might write it as "independence".

He is saying that other people who he perceived as more capable individuals ("more brilliant than" 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 ("invisible and despotic circles which delimit the universe of a certain milieu") 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, "If I have seen further it is by standing on the shoulders of Giants", 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 "Stand on the shoulders of giants" in recognition and tribute to this sentiment.

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    Good answer! Grothendieck was an earlier generation’s Perelman: exceptionally brilliant mathematician, and incapable to handle his own life with one fleeing to his mother’s basement, the other into a life as a farmer recluse. You don’t want life advice from a troubled person, so you’re really spot on. Feb 5 at 19:01
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    @gnometorule Indeed, if one hears from a world-famous athlete that the secret is "hard work and perseverance", it's fine to take those as necessary components of athletic success and motivation to keep to a rigorous routine, but foolish to take them as guaranteed sufficient for wealth and fame. Similarly, it seems worthwhile to take Grothendieck's advice that there is value in venturing outside mainstream thought to reach novel solutions, while shunning the notion that reclusiveness is the key to success.
    – Bryan Krause
    Feb 5 at 19:15
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    Ahem, the "standing on the shoulders of giants" is a double-edged argument. Newton used that to put down Hooke, who was of small build and regularly accused Newton of stealing his ideas. It sounds modest, but actually is quite a vicious (and funny) put-down. Feb 6 at 5:39
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    I think you read some things in his words that are not there. I don't think what he says is at odds with the sentiment of 'standing on the shoulders of giants'. Having dabbled in obscure math myself, it is a very lonely subject to begin with. But if you do try to do something genuinely new, you will literally have nobody to talk to it about; perhaps for decades. The reality is that it is only after your new ideas are established, anyone else will take the time to even consider them. Feb 6 at 8:52
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    Most, me included, dont have the stomach for that kind of chicken-egg dynamic, and would prefer to stay within the confines of more recognized domains. Its much psychologically easier to work on continuations of problems your peers worked on, which has associated conferences you can go to, and others to pat you on the back about, than it is to do something truly novel. All of that is true regardless of the respect you have for the shoulders you stand on. Nor do I see him advocating this path; he merely notes that he is that exceptional kind of loner, with zero regard for social approval. Feb 6 at 8:59
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All indications are that Grothendieck did not lack self-confidence. Just as well, considering his harsh and unlucky childhood and life situation prior to academic mathematics success.

It's not really that he "created a new subject" (modern algebraic geometry), but, rather, apparently found a path (that others were also seeking) to revivify the subject. He did have J.-P. Serre's support by correspondence, which not all of us do. :) Still, the homological ideas in his Tohoku J. article (in particular emphasizing sheaf cohomology as right-derived functors of the global-section functor) were not "entirely new", as Eilenberg and H. Cartan write their book at about the same time. As far as I know, though, the sheaf-cohomology idea was new, with Cech cohomology still a competitor into the 1960s.

A. Weil's 1949 "Foundations of Algebraic Geometry" was the main preceding attempt to rigorize (and extend...) the intuitive physical geometry of the Italian school from early 20th century. It did not seem to offer a good extension to deal with the Weil Conjectures, which Grothendieck had set his sights on.

He was evidently a forceful personality (I never met him) as well as a very good mathematician, and managed to persuade a small army of collaborators to work to implement/develop his vision.

At some point, evidently, he did become disenchanted with French academic culture... and "retired". To my mind, there is a considerable irony that the "establishment" he decided to disdain nevertheless supported him... and he apparently accepted most of that support.

Coming back to the specific question: I think it really amounts to a difference of taste. Grothendieck could not help himself but be rebellious, and disdained those who were not rebels. Oop, not counting the people who worked very hard to make his visions come to life? I myself am unsympathetic to this, but nevermind.

So, if we are to take it seriously, his claim was that those other talented people could have been as "great" as he, Grothendieck, if only they would have adopted his approach, instead of ... following their own?

At best, it seems to me that his remarks confound the institutions with the people.

EDIT: hoping to clarify, in response to some comments... First, I do not at all wish to diminish AG's rightfully earned reputation as one of the luminaries of the 20th century. But/and I would want to say that much of his legacy was, in fact, a team effort, with him as a/the leader. But, also, for sure, several or many other exceptional people "helped"! :) This fact is not as often made clear in the heroic legends. AG can be a heroic figure without diminishing the other very, very good mathematicians who made huge contributions (if not quite in his style).

Although I myself may be a small-time iconoclast, one thing I've thought a lot more about in the last 10-20 years is to try to give credit where credit is due. And not just to the "top ten" or similar most-heroic figures. And, factually, as my own scholariship has improved, I've learned that lots of great ideas were already manifest decades or a century or two earlier... but somehow not necessarily achieving top billing in textbooks and other professional-cultural mythology.

In AG's case, I think the Serre-Grothendieck correspondence shows the scholarly inputs Serre provided AG, although it is not easy to see that reflected in EGA nor SGA. The style of the latter did not seem much concerned with bibliographic/historical completeness.

I do think that one part of the Bourbaki impulse was an instance of "shedding the past"... partly for good reasons, but, also, throwing some babies out with the bathwater.

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    I am always surprised about the level of criticism that is leveled at nonconformists who go a path alone, with self-confidence, especially in a topic like mathematics where individuals actually can change a topic profoundly. I learned of course a lot from the "good citizens" in my time, but received the most impactful insights that affected my thinking most I from the "rebels" in our department. Today, the fact that these people become "the nail that stands out" seems to provoke the collective impulse to hammer them back to level. Not that they would stand any chance in today's climate anyway. Feb 6 at 5:48
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    Perhaps he resembled Godement? Godement made some remarks about some sources of funding, I think, in the introduction to Notes on Jacquet-Langlands Theory, and made other critical and provocative remarks besides.
    – Mehta
    Feb 6 at 10:41
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    Knowing how serious of a mathematician you are, I'm a bit surprised by how you seem to downplay the originality of G.'s work. There are so many new amazing ideas in the SGAs and in his work in the 80s, for instance. I don't see the irony either as a) he never returned to be a star in the math scene after leaving, even turning down the Crafoord prize, and b) his beef was with people he thought acted unethically, not the establishment intrinsically. Anyway I don't understand the "confounding institutions with people" - what does this quote have to do with institutions?
    – Cyclicduck
    Feb 7 at 5:40
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Many of the responses here give correct answers. They also explain that the view of this quote is not the norm, and that one shouldn't necessarily trust Grothendieck's advice based anyway. I agree with all of this, BUT with caveats so significant I believe it deserves a separate response.

Caveat 1: He is not implying that one should reject other people in the field and not cooperate with them.

As a matter of fact, in the same text he talks about how almost all his first ideas in algebraic geometry were stimulated by Serre. He was certainly no stranger to collaboration, and this is completely compatible with this quote. Rather than explaining abstractly this compatibility abstractly, it may be more explain to learn how he allegedly worked. In his IHES days, he would present his seminar once a week and follow it up with discussions. Sometimes he would also often invite others to his home to do math. But on other days, he would work alone, often late into the night, to come up with the notions he would discuss with other mathematicians.

The most well-known example of this is the notion of a scheme. This idea, or at least close variants of it, were up in the air at the time. But how did Grothendieck know that the very foundations of algebraic geometry should be rewritten with them? (a truly gargantuan task) According to this same text, he had to work alone, plumbing this concept himself to see this was the way to do it all. Another concept, the "Grothendieck topology" generalized the very notion of a topology in a seemingly naive way. But this was what was necessary to define etale cohomology, which led to the proof of the Weil conjectures.

For a more detailed discussion, see here. It seems to me that whenever he did math, he had a grand vision. This vision was certainly stimulated and enhanced by others, but it was something he could "see" on his own that was remarkably unconstrained.

Caveat 2: This quote is less a judgement of others than an expression of something personally important.

While this quote does indeed mention the impact of other mathematicians, I don't think that is the point of the paragraph. If you read on, you will find that this section is about the "interior adventure", especially about his experience of mathematics. He explains that for him, innocence and his ability to listen to the nature of things were the most essential traits to his success in math. While this formulation may sound romanticized, the content seems sound to me.

The notion that other mathematicians would have gained from what he describes is, in my opinion, a valid one. I believe most of the pushback against it comes from the fact that he is saying about himself, which may lead some people to associate it with arrogance. As a matter of fact, I have heard people talk about how various fields could have used a Grothendieck without any controversy.

Caveat 3: Just as one should not accept advice just because it is from Grothendieck, one should not reject it just because it is from Grothendieck.

There are some rather disparaging comments in the other answers about Grothendieck as a person, and how we should not listen to his advice as a result. I find this attitude very troubling for two reasons.

First, it is largely based on negative stereotypes that are absolutely false in this case. I've even seen many of the concrete facts about him be twisted or even made up to promote this sensationalist archetype of a recluse mathematician who understands nothing but math. Having read some of his later writings, I can confirm that such a depiction of Grothendieck is completely wrong. For sure, some of the events in his life did superficially resemble the stereotype, but his philosophy and understanding of life are absolutely different.

Second, one should in principle not discount somebody else's opinion based on the person, but judge it on its own merit. Just because someone was "troubled" doesn't mean that their view is worthless. (Especially when the cause of much of the trouble was a perception of the coming environmental crisis, leading him to be one of the first ecological activists.) In fact, as Grothendieck possessed both a unique and a supremely educated perspective, one should not be surprised if he has interesting things to say. At the very least, they will not be the same clichés that add no value.

Even in his writing outside of math, time and time again I have been amazed by how his perspective on a topic already accounted for my own. Experiencing this is certainly startling when you expect to be the one assessing the supposedly crazy views of another. His work is one of the very few that I have learned from upon reading multiple times, which is why I am very disappointed when people dismiss him based off of a stereotype.

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This is maybe unrelated to the question you are asking, but if your goal is to glean life lessons about how to be a good and successful researcher, then Grothendieck is probably a flawed source. Brilliant mathematician: Yes. Role model: Hardly.

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    Can you explain why, for the benefit of those of us who are not in the in-group of people who are experts on Grothendieck’s life and work?
    – Dan Romik
    Feb 6 at 5:04
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    I read that as "role model: Hardy" and was wondering why :-) To be honest, considering Grothendieck as a model for mathematician is like considering Feynman as a model for a physicist. It's impossible to copy that style, because one has to have the talent, and if one had it, one would and could go one's own way, anyway. Feb 6 at 5:51
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    @DanRomik I think it's fair to say that Grothendieck had mental health problems and led a lifestyle that would suit few of us. He resigned from his professor position at age 32 over political disagreements, largely withdrew from society, and spent the last 25 years of his life as a recluse in a mountain village, stateless and without citizenship. That's not a judgment, but it's also not a path through life that matches what many of us aspire to or dream about. Feb 6 at 20:33
  • @WolfgangBangerth thanks. To clarify, I wasn’t suspecting you of being judgmental, I just thought the answer cannot be understood without this type of background info about Grothendieck.
    – Dan Romik
    Feb 6 at 20:36
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    I'm disappointed that you say he's not a role model (in life), when he certainly is to many like me. Your facts are almost all incorrect - he resigned from the IHES at 42, not 32, he did not withdraw from society then but was in fact a normal professor till about 60, and he was a French citizen for the last half of his life - but that's beside the point. His commitment to his beliefs, his abundant generosity, the love he put into his work (not only in math) make him a unique role model for me.
    – Cyclicduck
    Feb 7 at 5:18
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Most of the great scientists and mathematicians work alone. The great Artist Pablo Picasso says that “Without great solitude, no serious work is possible.” And solitude refreshes body, mind and spirit.

Einstein says that the monotony and solitude of a quiet life stimulates the creative mind.

Aristotle says that the man who is content to live alone is either a beast or a god.

Pereleman also says that I decided to work alone and I had no closed friends. Most of his papers are not collaboration with others.

Grothendieck want to say that in alone your creativity and imagination power will increase. The more you are alone the more you have creativity power. This implies the meaning of capacity to be alone.

taken from this

Here is the outline of Grothendieck statement about alone

To state it in slightly different terms: in those critical years I learned how to be alone (*)

(*) This formulation doesn't really capture my meaning. I didn't, in any literal sense learn to be alone, for the simple reason that this knowledge had never been unlearned during my childhood. It is a basic capacity in all of us from the day of our birth. However these 3 years of work in isolation, when I was thrown onto my own resources, following guidelines which I myself had spontaneously invented, instilled in me a strong degree of confidence, unassuming yet enduring, in my ability to do mathematics, which owes nothing to any consensus or to the fashions which pass as law. I come back to this subject again in the note: "Roots and Solitude" ( R&S IV, #171.3, in particular page 1080). By this I mean to say: to reach out in my own way to the things I wished to learn, rather than relying on the notions of the consensus, overt or tacit, coming from a more or less extended clan of which I found myself a member, or which for any other reason laid claim to be taken as an authority. This silent consensus had informed me, both at the lyé and at the university, that one shouldn't bother worrying about what was really meant when using a term like "volume", which was "obviously self-evident", "generally known", "unproblematic", etc. I'd gone over their heads, almost as a matter of course, even as Lesbesgue himself had, several decades before, gone over their heads. It is in this gesture of "going beyond", to be something in oneself rather than the pawn of a consensus, the refusal to stay within a rigid circle that others have drawn around one - it is in this solitary act that one finds true creativity. All others things follow as a matter of course.

Grothendieck himself later wrote “I learned then, in solitude, the thing that is essential in the art of mathematics — that which no master can really teach” it is in this solitary act that one finds true creativity. All others things follow as a matter of course.

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    "Most of the great scientist and mathematician work alone." While that may once have been true (questionable), it certainly isn't today. Even we mathematicians swim in wide collaborative circles and have done for at least half a century. The internet has accelerated this, of course, since one needn't go on a pilgrimage to work with others outside the home institution. My first graduate experience (fifty or so years ago) was in a collaborative research seminar led by one of the "great minds".
    – Buffy
    Feb 6 at 13:44
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    @Buffy How can the majority be great? By definition the vast majority are within 2 standard deviations of the norm (assuming a normal distribution of skill, aptitude, etc.). I think Grothendieck is referring to those outliers far outside the norm.
    – M. Y. Zuo
    Feb 6 at 16:07
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    @Buffy I think the standard assumption when talking about characteristics of large groups of randomly selected people is to assume a normal distribution. Of course in reality the group of all scientists and mathematicians may in fact be fat-tailed, bimodal, etc., though I’ve not yet seen any convincing reason to assume so and in any case should certainly not be the default assumption. The other alternative of not assuming any distribution at all, i.e. perfect equality, also seems absurd.
    – M. Y. Zuo
    Feb 6 at 16:19
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    @Buffy, Right, I’m focusing on the distribution of published, and soon to be published, mathematicians and scientists, which is presumably what he’s referrring to. The population of even this select group, at least in modern times, is more than large enough for statistical reasoning.
    – M. Y. Zuo
    Feb 6 at 16:53
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    @MichaelZ, sorry, but you need to study statistics more deeply if you want to use it effectively.
    – Buffy
    Feb 6 at 17:28

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