Though the structures and patterns of mathematics reflect the structure of, and resonate in, the human mind every bit as much as do the structures and patterns of music, human beings have developed no mathematical equivalent to a pair of ears. Mathematics can only be "seen" with the "eyes of the mind". It is as if we had no sense of hearing, so that only someone able to sight read music would be able to appreciate its patterns and harmonies.
Keith DevlinHans Rosling typically would go into the room, and he would ask the audience questions. Often they had to answer them with clickers or raising their hands or something. We get [data] wrong because 50 years ago that wasn't the case and because we haven't had these graphics we don't realize that over the last 30, 40, 50 years things have changed dramatically. And you see how the world has been getting a better, safer, more homogeneous place. It just has.
Keith DevlinCardinal arithmetic will be quite important for us, so we spend some time on it. Since, however, it tends to be trivial, we shall not need to spend much of this time on proofs.
Keith DevlinMathematical thinking is not the same as doing mathematics - at least not as mathematics is typically presented in our school system. School math typically focuses on learning procedures to solve highly stereotyped problems. Professional mathematicians think a certain way to solve real problems, problems that can arise from the everyday world, or from science, or from within mathematics itself. The key to success in school math is to learn to think inside-the-box. In contrast, a key feature of mathematical thinking is thinking outside-the-box - a valuable ability in today's world.
Keith DevlinA PhD in Mathematics is three years of guessing it wrong, plus one week of getting it right and writing a dissertation.
Keith Devlin