The Princeton Companion to Mathematics, which came out just a few month ago, contains a wonderful short section entitled “Advice to a Young Mathematician” with advice from five eminent mathematicians. I was in the need of inspiration this weekend, and found some in these personal statements. Below the fold you will find a few excerpts applicable to any researcher of any age.
Readers: Please help me and other readers of this blog by posting in the comments section pointers to your favorite sources of research advice.
Sir Michael Atiyah: “My own approach has been to try to avoid the direct onslaught and look for indirect approaches. … it can lead to a beautiful and simple proof, which also “explains” why something is true. In fact, I believe the search for an explanation, for understanding, is what we should really be aiming for.”
Béla Bollobás: “Keep your ability to be surprised.”
Alain Connes: “Once a mathematician truly gets to know, in an original and “personal” manner, some small part of the mathematical world, however esoteric it may look at first, the journey can properly start.”
Dusa McDuff: “often one sees further by starting with the simplest questions and examples, because that makes it easier to understand the basic problem and then perhaps to find a new approach to it.”
Peter Sarnak: “Not to learn the tools is like trying to demolish a building with just a chisel. Even if you are very adept at using the chisel, somebody with a bulldozer will have a huge advantage and will not need to be nearly as skillful as you.”
I also loved this quote from Alain Connes: “Mathematicians usually have a hard time explaining to their partner that the times when they work with most intensity are when they are lying down in the dark on a sofa.” For me instead of “lying down in the dark,” it is “staring absently across the room.”
I’ll end with one final quote from Alain Connes that greatly amused me, but also struck me as largely true, at least with respect to certain areas of physics: “in general mathematicians tend to behave like “fermions,” i.e., they avoid working in areas that are too trendy, whereas physicists behave a lot more like “bosons,” which coalesce in large packs.”
There are other gems to be found in these short accounts; I may blog on some others in future posts. In the meantime, you can find the entire “Advice to a Young Mathematician” online, and please add comments with your sources of advice and inspiration.