The average student emerges at the end of the Ph.D. program, already middle-aged, overspecialized, poorly prepared for the world outside, and almost unemployable except in a narrow area of specialization. Large numbers of students for whom the program is inappropriate are trapped in it, because the Ph.D. has become a union card required for entry into the scientific job market.

The bottom line for mathematicians is that the architecture has to be right. In all the mathematics that I did, the essential point was to find the right architecture. It's like building a bridge. Once the main lines of the structure are right, then the details miraculously fit. The problem is the overall design.

I'm a mathematician, basically. What I do is look around for problems where I can find useful applications for mathematics. All I do, really, is the math, and other people have the ideas.

I think the biggest misconception about mathematics is that everybody has to learn it. That seems to be a complete mistake. All the time worrying about pushing the children and getting them to be mathematically literate and all that stuff. It's terribly hard on the kids. It's also hard on the teachers. And I think it's totally useless. To me, mathematics is like playing the violin. Some people can do it - others can't. If you don't have it, then there's no point in pretending.

I think that the artificial-intelligence people are making a lot of noise recently, claiming that artificial intelligence is making huge progress and we're going to be outstripped by the machines. But, in my view, this whole field is based on a misconception. I think the brain is analog, whereas the machines are digital. They really are different. So I think that what the machines can do, of course, is wonderful, but it's not the same as what the brain can do.

You could say science also is an art.

[On Richard P. Feynman's live demonstration of the rigidity of the O-rings when cold that doomed the space shuttle Challenger, killing seven astronauts:] The public saw with their own eyes how science is done, how a great scientist thinks with his hands, how nature gives a clear answer when a scientist asks her a clear question.