People think they don’t understand math, but it’s all about how you explain it to them. If you ask a drunkard what number is larger, 2/3 or 3/5, he won’t be able to tell you. But if you rephrase the question: what is better, 2 bottles of vodka for 3 people or 3 bottles of vodka for 5 people, he will tell you right away: 2 bottles for 3 people, of course.
People (or students) do not have shortcomings, only uniquenesses. The goal of a good teacher is to turn these uniquenesses into advantages.
Eugene Wigner wrote a famous essay on the unreasonable effectiveness of mathematics in natural sciences. He meant physics, of course. There is only one thing which is more unreasonable than the unreasonable effectiveness of mathematics in physics, and this is the unreasonable ineffectiveness of mathematics in biology.
One of the reasons we don't do as well as we should is that we are all over-taught.
I used to say: "Everything is Representation Theory". Now I say: "Nothing is Representation Theory".
The older I get, the more I believe that at the bottom of most deep mathematical problems there is a combinatorial problem.
If you understand something, you understand that it is obvious.
My life may be encapsulated by one of Graham Greene's "entertainments" titles: 'Loser Takes All'. Since I was thrown out of highschool for political reasons, I was free to study on my own and develop my own ways of thinking.
In the Middle Age, in Germany, if you wanted to learn addition and multiplication, you could go to any university. But if you wanted to learn division, you could only do it in one place, Heidelberg. This makes sense, since in my theory with Vladimir Retakh and Robert Wilson, addition and multiplications are cheap, but division is expensive.
My seminar is for highschool students, decent undergraduates, bright graduates, and outstanding professors.
They have the idea that non-commutative algebra should remind one of commutative algebra, but the former is more sophisticated. I believe that non-commutative algebra is just as simple, but it is different.
... The approach of von Neumann and Connes to the use of non-commutative algebra in physics is naive, the situation is much more complicated.
