Jordan Ellenberg makes a compelling case, as usual, on the pernicious cultural notion of “genius.” Jordan’s article also brought to mind a thought provoking piece on genius by Moon Duchin here (full disclosure: the link on Duchin’s website has the folowing caveat: note: juvenilia! Not that I hate it).
I should first clarify that I am essentially in agreement with many of Jordan’s points, as well as his implicit and explicit recommendations. That said, there is a secondary argument which portrays mathematics as a grand collective endeavour to which we can all contribute. I think that this is a little unrealistic. In my perspective, the actual number of people who are advancing mathematics in any genuine sense is very low. This is not to say that there aren’t quite a number of people doing *interesting* mathematics. But it’s not so clear the extent to which the discovery of conceptual breakthroughs is contingent on others first making incremental progress. This may sound like a depressing view of mathematics, but I don’t find it so. Merely to be an observer in the progress of number theory is enough for me — I know how to prove Fermat’s Last Theorem, how exciting is that*?
Having said this, there are two further points on which I do agree with Jordan. The first is that it is a terrible idea to *prejudge* who will be the ones to make progress, at least any more than necessary (judgements of various levels are completely built in to the academic world in the context of hiring, grants, etc.). This is consistent with Jordan’s point that one should view accomplishments rather than people as genius. The second is that the mere notion of genius has detrimental psychological effects on mathematicians. The depressing effect on enthusiasm for mathematics amongst students has already been mentioned, but I also wanted to note a contrasting effect. There’s a tendency amongst a certain group of students — for concreteness imagine a certain flavour of Harvard male undergraduate — who buys into the notion of genius with themselves as an example. This doesn’t do any favours for anyone; obviously not fellow students who are put off, but also the students themselves who are tempted to learn fancy machinery —- that’s what a genius does! — rather than to really understand the basics. Two clarifying points: first, this in no way applies to all Harvard undergraduates — some of the smartest I have met have been very humble and level headed. Second, when I single out Harvard undergraduates, I am implying a correlation rather than a causation: this is not at all the fault of the Harvard faculty who are generally no nonsense. Then again, Harvard does have Math 55, which is somewhat of a petri dish for this sort of attitude.
Having written this, I now see there is a practical problem in the classroom which I don’t know how to solve. What do you do when you are teaching and one student sticks out like a sore thumb because their understanding of the material (seemingly) far exceeds the rest of the class?
(*) OK, I have not read all the details of the proof of cyclic base change for \(\mathrm{GL}(2)\).
It’s not just conceptual breakthroughs (done by a few people) and incremental progress (towards big problems, I guess) that constitute the collective activity of mathematicians. There is also the working out of the consequences of the conceptual breakthroughs which may or may not be easy at the time and the few people who are responsible for the breakthroughs may not have time or interest in doing. Maybe another advance decades in the future will mean that those consequences can be worked out by a computer at the press of a button but having them done now means that we can all benefit from them.
First of all, thanks for the link to Duchin’s piece (which is really quite excellent) and for furthering the discussion about Jordan’s article. But I disagree on a number of levels, the most important of which is that you are not very precise about what exactly constitutes “advancing mathematics” (other than to implicitly equate it with creating a “conceptual breakthrough”), and it is my experience that a huge amount of what actually constitutes the advance of mathematics (in any meaningful norm) is made up of much, much smaller increments. These could be things like working out a nice example, or working it out in a nicer way; clarifying unclear but original thoughts of others; taking a result in one area and realizing its connections to other areas; doing computer experiments to support (or contradict) a conjecture; and so on. Another very important function of the vast ecosystem of mathematicians (as distinct from the lonely “few geniuses”) is that they play a vital role similar to the role of markets in setting prices in economies by the interplay of “supply and demand” – selectively working on or promoting the importance of some areas or problems rather than others (not to mention things like hiring, mentoring, etc.)
Now, some people are more original than others, or faster than others, or better at learning machinery, or better at translating between different domains of mathematics, or better algebraists, or better geometers, or better analysts, or have better taste, or have an unusual combination of talents etc. and these people make a disproportionately large contribution to mathematics (and are very often further credited with contributions that actually came from other people) but as big as this contribution is I think you overstate it, and understate the contribution of the hoi polloi. Maybe 10,000 Calegaris couldn’t replace a Gromov, but on the other hand, Gromov couldn’t replace 10,000 Calegaris either.
Perhaps it is more a matter of who is most easily *replaceable*. Consider the metaphor of building a cathederal: Gromov is the architect, and there are 10,000 Calegaris heaving limestone from a neighbouring quarry. Or perhaps, more generously, there are 100 Calegaris working as master stone masons. Whilst I might enjoy the chance to place a stone which could ultimately support a towering gothic column, It’s hard to argue that in my absence the same job could not ultimately have been done by someone else. (More realistically, I would be happy to spend my time sculpting a gargoyle which is a cross between a coelacanth and little Johnny Howard.) In this sense, nobody is really irreplaceable, but most are much easier to replace than others.
As someone who could multiply two digit numbers at the age of 5 but grew up in the rural midwest, I can’t help but feel a little resentment that JSE thinks we should pay less attention to “gifted” students.
I didn’t get the impression that JSE would be in any way against gifted children training programmes, but I can let him answer for himself. As for the best way to handle prodigies, I can’t really answer that (growing up pretty much average amongst my siblings).
To KV: Yeah, I was worried that the article would be read as saying “We shouldn’t have gifted programs” or denigrating the people who work really hard to build things like Canada/USA MathCamp or Proof School or the rather amazing Young Shakespeare Theater here in Madison. After all, I benefited greatly from a well-funded gifted education program for slightly older students called “Harvard College.”
On the other hand, I am concerned that the money and resources aimed at gifted-identified students will exactly FAIL to flow to the rural Midwest, low-income communities in the cities, etc. Those are the kids who need it more than my kids ever will. I can teach my kids advanced math myself, should they have a taste for it.
Just to take number theory, I dunno, does the “irreplaceable few” theory really applies outside of certain very specialized areas?
Examples — primes in P, Catalan’s conjecture, gaps between primes — if you look at the breakthroughs in those cases, I don’t think you could have guessed where they were coming from beforehand. I tend to think that for such problems it really was a matter of a lot of people thinking about it, each with a small probability of success.
It’s certainly true that in certain areas it seems that a few people are pushing all the progress. It also often seems to be true that these areas have very high entry barriers — it takes many years of study to even understand the problem statements, and of the few who are willing to invest so much time, even fewer have access to all the tools really necessary to seriously attack them (because you have to learn those things in person, I think).
Ah, but I’m not claiming that one can predict before hand who will produce the breakthroughs, merely that the genuine breakthroughs will come from a small minority of mathematicians.
Is this not in some danger of being true by definition – because breakthroughs are by definition rare (if a reasonable proportion of mathematicians were producing breakthroughs, they would become commonplace, and thus not breakthroughs)?
In some sense you are correct, so I need to strengthen my claim slightly. Do you think that anything that you or I do will have any lasting impact (even over a small a period of, say, 30 years) in the Langlands programme? Answering for myself, I think the answer quite possibly might be “no,” and I would be perfectly comfortable with that, because I think that the number of people for whom the answer would be yes will be pretty small. In other words, one problem with using the term “breakthrough” is that it could be interpreted as “an uncommonly good idea,” so instead one should interpret it as “an idea that still has relevance in thirty years…”
Of course, all of this may simply be evidence that *I* cannot escape from the cultural notion of “genius”…
(I don’t think I can take the reply any deeper, so hopefully this appears in a sensible place.)
For myself, the chance of having a lasting impact over 30 years is miniscule, and probably the same is true over even 10 years, though hopefully not over 5. (I guess BLGGT is 4 years old and not yet on the scrap heap.) I’m certainly completely comfortable with this, and indeed I’m essentially in agreement with you on all of this.
so instead one should interpret it as “an idea that still has relevance in thirty years…”
But my impression is that when one does serious intellectual history, one realizes that ideas with such great historical relevance are usually the product of a community rather than the work of a particular individual, and even when one individual made single-handedly an enormous contribution, it seems probable that the idea would have surfaced anyway (usually not long thereafter) absent this contribution. “Progress […] obtained naturally and cumulatively as a consequence of hard work, directed by intuition, literature, and a bit of luck” seems to be the rule all the way down.
Langlands program does seem to be close to an ideal example for your thesis, though, but it could be a rather isolated example indeed.