Vanity and Ambition in Mathematics

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In my time in the mathematical community I've formed the subjective impression that it's noticeably less common for mathematicians of the highest caliber to engage in status games than members of the general population do. This impression is consistent with the modesty that comes across in the writings of such mathematicians. I record some relevant quotations below and then discuss interpretations of the situation.

Acknowledgment - I learned of the Hironaka interview quoted below from my colleague Laurens Gunnarsen.

Edited 10/12/10 to remove the first portion of the Hironaka quote which didn't capture the phenomenon that I'm trying to get at here.

In a 2005 Interview for the Notices of the AMS, one of the reasons that Fields Medalist Heisuke Hironaka says

By the way, Mori is a genius. I am not. So that is a big difference! Mori was a student when I was a visiting professor at Kyoto University. I gave lectures in Kyoto, and Mori wrote notes, which were published in a book. He was really amazing. My lectures were terrible, but when I looked at his notes, it was all there! Mori is a discoverer. He finds new things that people never imagined.

(I'll note in passing that the sense of the "genius" that Hironaka is using here is probably different than the sense of "genius" that Gowers uses in Mathematics: A Very Short Introduction.)

In his review of Haruzo Hida’s p-adic automorphic forms on Shimura varieties the originator of the Langlands program Robert Langlands wrote

So ill-equipped as I am in many ways – although not in all – my first, indeed my major task was to take bearings. The second is, bearings taken, doubtful or not, to communicate them at least to an experienced reader and, in so far as this is possible, even to an inexperienced one. For lack of time and competence I accomplished neither task satisfactorily. So, although I have made a real effort, this review is not the brief, limpid yet comprehensive, account of the subject, revealing its manifold possibilities, that I would have liked to write and that it deserves. The review is imbalanced and there is too much that I had to leave obscure, too many possibly premature intimations. A reviewer with greater competence, who saw the domain whole and, in addition, had a command of the detail would have done much better.

For context, it's worthwhile to note that Langlands' own work is used in an essential way in Hida's book.

The 2009 Abel Prize Interview with Mikhail Gromov contains the following questions and answers:

Raussen and Skau: Can you remember when and how you became aware of your exceptional mathematical talent?

Gromov: I do not think I am exceptional. Accidentally, things happened, and I have qualities that you can appreciate. I guess I never thought in those terms.

[...]

Raussen and Skau: Is there one particular theorem or result you are the most proud of?

Gromov: Yes. It is my introduction of pseudoholomorphic curves, unquestionably. Everything else was just understanding what was already known and to make it look like a new kind of discovery.

In his MathOverflow self-summary, William Thurston wrote

Mathematics is a process of staring hard enough with enough perseverance at at the fog of muddle and confusion to eventually break through to improved clarity. I'm happy when I can admit, at least to myself, that my thinking is muddled, and I try to overcome the embarrassment that I might reveal ignorance or confusion. Over the years, this has helped me develop clarity in some things, but I remain muddled in many others. I enjoy questions that seem honest, even when they admit or reveal confusion, in preference to questions that appear designed to project sophistication.


I interpret the above quotations (and many others by similar such people) to point to a markedly lower than usual interest in status. As JoshuaZ points out, one could instead read the quotations as counter-signaling, but such an interpretation feels like a stretch to me. I doubt that in practice such remarks serve as an effective counter-signal. More to the point, there's a compelling alternate explanation for why one would see lower than usual levels of status signaling among mathematicians of the highest caliber. Gromov hints at this in the aforementioned interview:

Raussen and Skau: We are surprised that you are so modest by playing down your own achievements. Maybe your ideas are naíve, as you yourself say; but to get results from these ideas, that requires some ingenuity, doesn’t it?

Gromov: It is not that I am terribly modest. I don’t think I am a complete idiot. Typically when you do mathematics you don’t think about yourself. A friend of mine was complaining that anytime he had a good idea he became so excited about how smart he was that he could not work afterwards. So naturally, I try not to think about it.

In Récoltes et Semailles, Alexander Grothendieck offered a more detailed explanation:

The truth of the matter is that it is universally the case that, in the real motives of the scientist, of which he himself is often unaware in his work, vanity and ambition will play as large a role as they do in all other professions. The forms that these assume can be in turn subtle or grotesque, depending on the individual. Nor do I exempt myself. Anyone who reads this testimonial will have to agree with me.

It is also the case that the most totally consuming ambition is powerless to make or to demonstrate the simplest mathematical discovery - even as it is powerless (for example) to "score" (in the vulgar sense). Whether one is male or female, that which allows one to 'score' is not ambition, the desire to shine, to exhibit one's prowess, sexual in this case. Quite the contrary!

What brings success in this case is the acute perception of the presence of something strong, very real and at the same time very delicate. Perhaps one can call it "beauty", in its thousand-fold aspects. That someone is ambitious doesn't mean that one cannot also feel the presence of beauty in them; but it is not the attribute of ambition which evokes this feeling....

The first man to discover and master fire was just like you and me. He was neither a hero nor a demi-god. Once again like you and me he had experienced the sting of anguish, and applied the poultice of vanity to anaesthetize that sting. But, at the moment at which he first "knew" fire he had neither fear nor vanity. That is the truth at the heart of all heroic myth. The myth itself becomes insipid, nothing but a drug, when it is used to conceal the true nature of things.

[...]

In our acquisition of knowledge of the Universe (whether mathematical or otherwise) that which renovates the quest is nothing more nor less than complete innocence. It is in this state of complete innocence that we receive everything from the moment of our birth. Although so often the object of our contempt and of our private fears, it is always in us. It alone can unite humility with boldness so as to allow us to penetrate to the heart of things, or allow things to enter us and taken possession of us.

This unique power is in no way a privilege given to "exceptional talents" - persons of incredible brain power (for example), who are better able to manipulate, with dexterity and ease, an enormous mass of data, ideas and specialized skills. Such gifts are undeniably valuable, and certainly worthy of envy from those who (like myself) were not so endowed at birth," far beyond the ordinary".

Yet it is not these gifts, nor the most determined ambition combined with irresistible will-power, that enables one to surmount the "invisible yet formidable boundaries" that encircle our universe. Only innocence can surmount them, which mere knowledge doesn't even take into account, in those moments when we find ourselves able to listen to things, totally and intensely absorbed in child play.

The amount of focus on the subject itself which is required to do mathematical research of the highest caliber is very high. It's plausible that the focuses entailed by vanity and ambition are detrimental to subject matter focus. If this is true (as I strongly suspect to be the case based on my own experience, my observations of others, the remarks of colleagues, and the remarks of eminent figures like Gromov and Grothendieck), aspiring mathematicians would do well to work to curb their ambition and vanity and increase their attraction to mathematics for its own sake.

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