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 .