I took the liberty of reproducing some sections of this rather long interview: it was a pleasure to hear their views on matters that we all wrestle with.
Any good theorem should have several proofs, the more the better. For two reasons: usually, different proofs have different strengths and weaknesses, and they generalize in different directions - they are not just repetitions of each other. And that is certainly the case with the proofs that we came up with. There are different reasons for the proofs, they have different histories and backgrounds. Some of them are good for this application, some are good for that application. They all shed light on the area. If you cannot look at a problem from different directions, it is probably not very interesting; the more perspectives, the better!On the specialization in math: this has been a topic of discussion in theory as well, and their holistic view of mathematics is comforting to those of us who see the inevitable splitting of the subareas of theoretical computer science. It also reminds me of Avi Wigderson's lecture at STOC this year on the way different areas in theory are connected.
It is artificial to divide mathematics into separate chunks, and then to say that you bring them together as though this is a surprise. On the contrary, they are all part of the puzzle of mathematics. Sometimes you would develop some things for their own sake for a while e.g. if you develop group theory by itself. But that is just a sort of temporary convenient division of labour. Fundamentally, mathematics should be used as a unity. I think the more examples we have of people showing that you can usefully apply analysis to geometry, the better. And not just analysis, I think that some physics came into it as well: Many of the ideas in geometry use physical insight as well - take the example of Riemann! This is all part of the broad mathematical tradition, which sometimes is in danger of being overlooked by modern, younger people who say "we have separate divisions". We do not want to have any of that kind, really.On why researchers tend to get specialized (too) quickly:
In the United States I observe a trend towards early specialization driven by economic considerations. You must show early promise to get good letters of recommendations to get good first jobs. You can't afford to branch out until you have established yourself and have a secure position. The realities of life force a narrowness in perspective that is not inherent to mathematics. We can counter too much specialization with new resources that would give young people more freedom than they presently have, freedom to explore mathematics more broadly, or to explore connections with other subjects, like biology these day where there is lots to be discovered.And finally, an eloquent argument for simplicity in proofs:
The passing of mathematics on to subsequent generations is essential for the future, and this is only possible if every generation of mathematicians understands what they are doing and distils it out in such a form that it is easily understood by the next generation. Many complicated things get simple when you have the right point of view. The first proof of something may be very complicated, but when you understand it well, you readdress it, and eventually you can present it in a way that makes it look much more understandable - and that's the way you pass it on to the next generation! Without that, we could never make progress - we would have all this messy stuff.