For most of us, all we can hope for is to become “a first rate–second rate mathematician.” This is already a lofty goal. There are just a tiny number of first rate mathematicians. Luckily, an army can’t move forward if it consists only of generals. It takes a broad spectrum of mathematicians with all kinds of different talents to propel the subject forward. Also, the most critical need in mathematics is for truly creative ideas—and these can come from anyone.
When asked once how his day was going, the cartoon figure Charlie Brown responded:I keep hoping yesterday will be better.
Under this system, we should hand someone a Fields Medal for being the best and the brightest of the younger generation and say: “Here is your medal. By the way, your career is over.”
A story in the mathematics community is used to explain the difference between a mathematician and a physicist. Physicists work for ten years on a difficult problem and when they are done they say: “I am a genius for figuring this out.” Mathematicians work for ten years on a problem and when they are done they say: “I am an idiot. The answer was obvious.”
A famous review of a published math paper reads: The results in this paper are false. The mistakes are not new.
The difference between genius and stupidity is that genius has its limits.